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INSTRUCTORS Raffi Hovasapian John Zhu
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Raffi Hovasapian

Raffi Hovasapian

Integration by Parts

Slide Duration:

Table of Contents

I. Limits and Derivatives
Overview & Slopes of Curves

42m 8s

Intro
0:00
Overview & Slopes of Curves
0:21
Differential and Integral
0:22
Fundamental Theorem of Calculus
6:36
Differentiation or Taking the Derivative
14:24
What Does the Derivative Mean and How do We Find it?
15:18
Example: f'(x)
19:24
Example: f(x) = sin (x)
29:16
General Procedure for Finding the Derivative of f(x)
37:33
More on Slopes of Curves

50m 53s

Intro
0:00
Slope of the Secant Line along a Curve
0:12
Slope of the Tangent Line to f(x) at a Particlar Point
0:13
Slope of the Secant Line along a Curve
2:59
Instantaneous Slope
6:51
Instantaneous Slope
6:52
Example: Distance, Time, Velocity
13:32
Instantaneous Slope and Average Slope
25:42
Slope & Rate of Change
29:55
Slope & Rate of Change
29:56
Example: Slope = 2
33:16
Example: Slope = 4/3
34:32
Example: Slope = 4 (m/s)
39:12
Example: Density = Mass / Volume
40:33
Average Slope, Average Rate of Change, Instantaneous Slope, and Instantaneous Rate of Change
47:46
Example Problems for Slopes of Curves

59m 12s

Intro
0:00
Example I: Water Tank
0:13
Part A: Which is the Independent Variable and Which is the Dependent?
2:00
Part B: Average Slope
3:18
Part C: Express These Slopes as Rates-of-Change
9:28
Part D: Instantaneous Slope
14:54
Example II: y = √(x-3)
28:26
Part A: Calculate the Slope of the Secant Line
30:39
Part B: Instantaneous Slope
41:26
Part C: Equation for the Tangent Line
43:59
Example III: Object in the Air
49:37
Part A: Average Velocity
50:37
Part B: Instantaneous Velocity
55:30
Desmos Tutorial

18m 43s

Intro
0:00
Desmos Tutorial
1:42
Desmos Tutorial
1:43
Things You Must Learn To Do on Your Particular Calculator
2:39
Things You Must Learn To Do on Your Particular Calculator
2:40
Example I: y=sin x
4:54
Example II: y=x³ and y = d/(dx) (x³)
9:22
Example III: y = x² {-5 <= x <= 0} and y = cos x {0 < x < 6}
13:15
The Limit of a Function

51m 53s

Intro
0:00
The Limit of a Function
0:14
The Limit of a Function
0:15
Graph: Limit of a Function
12:24
Table of Values
16:02
lim x→a f(x) Does not Say What Happens When x = a
20:05
Example I: f(x) = x²
24:34
Example II: f(x) = 7
27:05
Example III: f(x) = 4.5
30:33
Example IV: f(x) = 1/x
34:03
Example V: f(x) = 1/x²
36:43
The Limit of a Function, Cont.
38:16
Infinity and Negative Infinity
38:17
Does Not Exist
42:45
Summary
46:48
Example Problems for the Limit of a Function

24m 43s

Intro
0:00
Example I: Explain in Words What the Following Symbols Mean
0:10
Example II: Find the Following Limit
5:21
Example III: Use the Graph to Find the Following Limits
7:35
Example IV: Use the Graph to Find the Following Limits
11:48
Example V: Sketch the Graph of a Function that Satisfies the Following Properties
15:25
Example VI: Find the Following Limit
18:44
Example VII: Find the Following Limit
20:06
Calculating Limits Mathematically

53m 48s

Intro
0:00
Plug-in Procedure
0:09
Plug-in Procedure
0:10
Limit Laws
9:14
Limit Law 1
10:05
Limit Law 2
10:54
Limit Law 3
11:28
Limit Law 4
11:54
Limit Law 5
12:24
Limit Law 6
13:14
Limit Law 7
14:38
Plug-in Procedure, Cont.
16:35
Plug-in Procedure, Cont.
16:36
Example I: Calculating Limits Mathematically
20:50
Example II: Calculating Limits Mathematically
27:37
Example III: Calculating Limits Mathematically
31:42
Example IV: Calculating Limits Mathematically
35:36
Example V: Calculating Limits Mathematically
40:58
Limits Theorem
44:45
Limits Theorem 1
44:46
Limits Theorem 2: Squeeze Theorem
46:34
Example VI: Calculating Limits Mathematically
49:26
Example Problems for Calculating Limits Mathematically

21m 22s

Intro
0:00
Example I: Evaluate the Following Limit by Showing Each Application of a Limit Law
0:16
Example II: Evaluate the Following Limit
1:51
Example III: Evaluate the Following Limit
3:36
Example IV: Evaluate the Following Limit
8:56
Example V: Evaluate the Following Limit
11:19
Example VI: Calculating Limits Mathematically
13:19
Example VII: Calculating Limits Mathematically
14:59
Calculating Limits as x Goes to Infinity

50m 1s

Intro
0:00
Limit as x Goes to Infinity
0:14
Limit as x Goes to Infinity
0:15
Let's Look at f(x) = 1 / (x-3)
1:04
Summary
9:34
Example I: Calculating Limits as x Goes to Infinity
12:16
Example II: Calculating Limits as x Goes to Infinity
21:22
Example III: Calculating Limits as x Goes to Infinity
24:10
Example IV: Calculating Limits as x Goes to Infinity
36:00
Example Problems for Limits at Infinity

36m 31s

Intro
0:00
Example I: Calculating Limits as x Goes to Infinity
0:14
Example II: Calculating Limits as x Goes to Infinity
3:27
Example III: Calculating Limits as x Goes to Infinity
8:11
Example IV: Calculating Limits as x Goes to Infinity
14:20
Example V: Calculating Limits as x Goes to Infinity
20:07
Example VI: Calculating Limits as x Goes to Infinity
23:36
Continuity

53m

Intro
0:00
Definition of Continuity
0:08
Definition of Continuity
0:09
Example: Not Continuous
3:52
Example: Continuous
4:58
Example: Not Continuous
5:52
Procedure for Finding Continuity
9:45
Law of Continuity
13:44
Law of Continuity
13:45
Example I: Determining Continuity on a Graph
15:55
Example II: Show Continuity & Determine the Interval Over Which the Function is Continuous
17:57
Example III: Is the Following Function Continuous at the Given Point?
22:42
Theorem for Composite Functions
25:28
Theorem for Composite Functions
25:29
Example IV: Is cos(x³ + ln x) Continuous at x=π/2?
27:00
Example V: What Value of A Will make the Following Function Continuous at Every Point of Its Domain?
34:04
Types of Discontinuity
39:18
Removable Discontinuity
39:33
Jump Discontinuity
40:06
Infinite Discontinuity
40:32
Intermediate Value Theorem
40:58
Intermediate Value Theorem: Hypothesis & Conclusion
40:59
Intermediate Value Theorem: Graphically
43:40
Example VI: Prove That the Following Function Has at Least One Real Root in the Interval [4,6]
47:46
Derivative I

40m 2s

Intro
0:00
Derivative
0:09
Derivative
0:10
Example I: Find the Derivative of f(x)=x³
2:20
Notations for the Derivative
7:32
Notations for the Derivative
7:33
Derivative & Rate of Change
11:14
Recall the Rate of Change
11:15
Instantaneous Rate of Change
17:04
Graphing f(x) and f'(x)
19:10
Example II: Find the Derivative of x⁴ - x²
24:00
Example III: Find the Derivative of f(x)=√x
30:51
Derivatives II

53m 45s

Intro
0:00
Example I: Find the Derivative of (2+x)/(3-x)
0:18
Derivatives II
9:02
f(x) is Differentiable if f'(x) Exists
9:03
Recall: For a Limit to Exist, Both Left Hand and Right Hand Limits Must Equal to Each Other
17:19
Geometrically: Differentiability Means the Graph is Smooth
18:44
Example II: Show Analytically that f(x) = |x| is Nor Differentiable at x=0
20:53
Example II: For x > 0
23:53
Example II: For x < 0
25:36
Example II: What is f(0) and What is the lim |x| as x→0?
30:46
Differentiability & Continuity
34:22
Differentiability & Continuity
34:23
How Can a Function Not be Differentiable at a Point?
39:38
How Can a Function Not be Differentiable at a Point?
39:39
Higher Derivatives
41:58
Higher Derivatives
41:59
Derivative Operator
45:12
Example III: Find (dy)/(dx) & (d²y)/(dx²) for y = x³
49:29
More Example Problems for The Derivative

31m 38s

Intro
0:00
Example I: Sketch f'(x)
0:10
Example II: Sketch f'(x)
2:14
Example III: Find the Derivative of the Following Function sing the Definition
3:49
Example IV: Determine f, f', and f'' on a Graph
12:43
Example V: Find an Equation for the Tangent Line to the Graph of the Following Function at the Given x-value
13:40
Example VI: Distance vs. Time
20:15
Example VII: Displacement, Velocity, and Acceleration
23:56
Example VIII: Graph the Displacement Function
28:20
II. Differentiation
Differentiation of Polynomials & Exponential Functions

47m 35s

Intro
0:00
Differentiation of Polynomials & Exponential Functions
0:15
Derivative of a Function
0:16
Derivative of a Constant
2:35
Power Rule
3:08
If C is a Constant
4:19
Sum Rule
5:22
Exponential Functions
6:26
Example I: Differentiate
7:45
Example II: Differentiate
12:38
Example III: Differentiate
15:13
Example IV: Differentiate
16:20
Example V: Differentiate
19:19
Example VI: Find the Equation of the Tangent Line to a Function at a Given Point
12:18
Example VII: Find the First & Second Derivatives
25:59
Example VIII
27:47
Part A: Find the Velocity & Acceleration Functions as Functions of t
27:48
Part B: Find the Acceleration after 3 Seconds
30:12
Part C: Find the Acceleration when the Velocity is 0
30:53
Part D: Graph the Position, Velocity, & Acceleration Graphs
32:50
Example IX: Find a Cubic Function Whose Graph has Horizontal Tangents
34:53
Example X: Find a Point on a Graph
42:31
The Product, Power & Quotient Rules

47m 25s

Intro
0:00
The Product, Power and Quotient Rules
0:19
Differentiate Functions
0:20
Product Rule
5:30
Quotient Rule
9:15
Power Rule
10:00
Example I: Product Rule
13:48
Example II: Quotient Rule
16:13
Example III: Power Rule
18:28
Example IV: Find dy/dx
19:57
Example V: Find dy/dx
24:53
Example VI: Find dy/dx
28:38
Example VII: Find an Equation for the Tangent to the Curve
34:54
Example VIII: Find d²y/dx²
38:08
Derivatives of the Trigonometric Functions

41m 8s

Intro
0:00
Derivatives of the Trigonometric Functions
0:09
Let's Find the Derivative of f(x) = sin x
0:10
Important Limits to Know
4:59
d/dx (sin x)
6:06
d/dx (cos x)
6:38
d/dx (tan x)
6:50
d/dx (csc x)
7:02
d/dx (sec x)
7:15
d/dx (cot x)
7:27
Example I: Differentiate f(x) = x² - 4 cos x
7:56
Example II: Differentiate f(x) = x⁵ tan x
9:04
Example III: Differentiate f(x) = (cos x) / (3 + sin x)
10:56
Example IV: Differentiate f(x) = e^x / (tan x - sec x)
14:06
Example V: Differentiate f(x) = (csc x - 4) / (cot x)
15:37
Example VI: Find an Equation of the Tangent Line
21:48
Example VII: For What Values of x Does the Graph of the Function x + 3 cos x Have a Horizontal Tangent?
25:17
Example VIII: Ladder Problem
28:23
Example IX: Evaluate
33:22
Example X: Evaluate
36:38
The Chain Rule

24m 56s

Intro
0:00
The Chain Rule
0:13
Recall the Composite Functions
0:14
Derivatives of Composite Functions
1:34
Example I: Identify f(x) and g(x) and Differentiate
6:41
Example II: Identify f(x) and g(x) and Differentiate
9:47
Example III: Differentiate
11:03
Example IV: Differentiate f(x) = -5 / (x² + 3)³
12:15
Example V: Differentiate f(x) = cos(x² + c²)
14:35
Example VI: Differentiate f(x) = cos⁴x +c²
15:41
Example VII: Differentiate
17:03
Example VIII: Differentiate f(x) = sin(tan x²)
19:01
Example IX: Differentiate f(x) = sin(tan² x)
21:02
More Chain Rule Example Problems

25m 32s

Intro
0:00
Example I: Differentiate f(x) = sin(cos(tanx))
0:38
Example II: Find an Equation for the Line Tangent to the Given Curve at the Given Point
2:25
Example III: F(x) = f(g(x)), Find F' (6)
4:22
Example IV: Differentiate & Graph both the Function & the Derivative in the Same Window
5:35
Example V: Differentiate f(x) = ( (x-8)/(x+3) )⁴
10:18
Example VI: Differentiate f(x) = sec²(12x)
12:28
Example VII: Differentiate
14:41
Example VIII: Differentiate
19:25
Example IX: Find an Expression for the Rate of Change of the Volume of the Balloon with Respect to Time
21:13
Implicit Differentiation

52m 31s

Intro
0:00
Implicit Differentiation
0:09
Implicit Differentiation
0:10
Example I: Find (dy)/(dx) by both Implicit Differentiation and Solving Explicitly for y
12:15
Example II: Find (dy)/(dx) of x³ + x²y + 7y² = 14
19:18
Example III: Find (dy)/(dx) of x³y² + y³x² = 4x
21:43
Example IV: Find (dy)/(dx) of the Following Equation
24:13
Example V: Find (dy)/(dx) of 6sin x cos y = 1
29:00
Example VI: Find (dy)/(dx) of x² cos² y + y sin x = 2sin x cos y
31:02
Example VII: Find (dy)/(dx) of √(xy) = 7 + y²e^x
37:36
Example VIII: Find (dy)/(dx) of 4(x²+y²)² = 35(x²-y²)
41:03
Example IX: Find (d²y)/(dx²) of x² + y² = 25
44:05
Example X: Find (d²y)/(dx²) of sin x + cos y = sin(2x)
47:48
III. Applications of the Derivative
Linear Approximations & Differentials

47m 34s

Intro
0:00
Linear Approximations & Differentials
0:09
Linear Approximations & Differentials
0:10
Example I: Linear Approximations & Differentials
11:27
Example II: Linear Approximations & Differentials
20:19
Differentials
30:32
Differentials
30:33
Example III: Linear Approximations & Differentials
34:09
Example IV: Linear Approximations & Differentials
35:57
Example V: Relative Error
38:46
Related Rates

45m 33s

Intro
0:00
Related Rates
0:08
Strategy for Solving Related Rates Problems #1
0:09
Strategy for Solving Related Rates Problems #2
1:46
Strategy for Solving Related Rates Problems #3
2:06
Strategy for Solving Related Rates Problems #4
2:50
Strategy for Solving Related Rates Problems #5
3:38
Example I: Radius of a Balloon
5:15
Example II: Ladder
12:52
Example III: Water Tank
19:08
Example IV: Distance between Two Cars
29:27
Example V: Line-of-Sight
36:20
More Related Rates Examples

37m 17s

Intro
0:00
Example I: Shadow
0:14
Example II: Particle
4:45
Example III: Water Level
10:28
Example IV: Clock
20:47
Example V: Distance between a House and a Plane
29:11
Maximum & Minimum Values of a Function

40m 44s

Intro
0:00
Maximum & Minimum Values of a Function, Part 1
0:23
Absolute Maximum
2:20
Absolute Minimum
2:52
Local Maximum
3:38
Local Minimum
4:26
Maximum & Minimum Values of a Function, Part 2
6:11
Function with Absolute Minimum but No Absolute Max, Local Max, and Local Min
7:18
Function with Local Max & Min but No Absolute Max & Min
8:48
Formal Definitions
10:43
Absolute Maximum
11:18
Absolute Minimum
12:57
Local Maximum
14:37
Local Minimum
16:25
Extreme Value Theorem
18:08
Theorem: f'(c) = 0
24:40
Critical Number (Critical Value)
26:14
Procedure for Finding the Critical Values of f(x)
28:32
Example I: Find the Critical Values of f(x) x + sinx
29:51
Example II: What are the Absolute Max & Absolute Minimum of f(x) = x + 4 sinx on [0,2π]
35:31
Example Problems for Max & Min

40m 44s

Intro
0:00
Example I: Identify Absolute and Local Max & Min on the Following Graph
0:11
Example II: Sketch the Graph of a Continuous Function
3:11
Example III: Sketch the Following Graphs
4:40
Example IV: Find the Critical Values of f (x) = 3x⁴ - 7x³ + 4x²
6:13
Example V: Find the Critical Values of f(x) = |2x - 5|
8:42
Example VI: Find the Critical Values
11:42
Example VII: Find the Critical Values f(x) = cos²(2x) on [0,2π]
16:57
Example VIII: Find the Absolute Max & Min f(x) = 2sinx + 2cos x on [0,(π/3)]
20:08
Example IX: Find the Absolute Max & Min f(x) = (ln(2x)) / x on [1,3]
24:39
The Mean Value Theorem

25m 54s

Intro
0:00
Rolle's Theorem
0:08
Rolle's Theorem: If & Then
0:09
Rolle's Theorem: Geometrically
2:06
There May Be More than 1 c Such That f'( c ) = 0
3:30
Example I: Rolle's Theorem
4:58
The Mean Value Theorem
9:12
The Mean Value Theorem: If & Then
9:13
The Mean Value Theorem: Geometrically
11:07
Example II: Mean Value Theorem
13:43
Example III: Mean Value Theorem
21:19
Using Derivatives to Graph Functions, Part I

25m 54s

Intro
0:00
Using Derivatives to Graph Functions, Part I
0:12
Increasing/ Decreasing Test
0:13
Example I: Find the Intervals Over Which the Function is Increasing & Decreasing
3:26
Example II: Find the Local Maxima & Minima of the Function
19:18
Example III: Find the Local Maxima & Minima of the Function
31:39
Using Derivatives to Graph Functions, Part II

44m 58s

Intro
0:00
Using Derivatives to Graph Functions, Part II
0:13
Concave Up & Concave Down
0:14
What Does This Mean in Terms of the Derivative?
6:14
Point of Inflection
8:52
Example I: Graph the Function
13:18
Example II: Function x⁴ - 5x²
19:03
Intervals of Increase & Decrease
19:04
Local Maxes and Mins
25:01
Intervals of Concavity & X-Values for the Points of Inflection
29:18
Intervals of Concavity & Y-Values for the Points of Inflection
34:18
Graphing the Function
40:52
Example Problems I

49m 19s

Intro
0:00
Example I: Intervals, Local Maxes & Mins
0:26
Example II: Intervals, Local Maxes & Mins
5:05
Example III: Intervals, Local Maxes & Mins, and Inflection Points
13:40
Example IV: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity
23:02
Example V: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity
34:36
Example Problems III

59m 1s

Intro
0:00
Example I: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
0:11
Example II: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
21:24
Example III: Cubic Equation f(x) = Ax³ + Bx² + Cx + D
37:56
Example IV: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
46:19
L'Hospital's Rule

30m 9s

Intro
0:00
L'Hospital's Rule
0:19
Indeterminate Forms
0:20
L'Hospital's Rule
3:38
Example I: Evaluate the Following Limit Using L'Hospital's Rule
8:50
Example II: Evaluate the Following Limit Using L'Hospital's Rule
10:30
Indeterminate Products
11:54
Indeterminate Products
11:55
Example III: L'Hospital's Rule & Indeterminate Products
13:57
Indeterminate Differences
17:00
Indeterminate Differences
17:01
Example IV: L'Hospital's Rule & Indeterminate Differences
18:57
Indeterminate Powers
22:20
Indeterminate Powers
22:21
Example V: L'Hospital's Rule & Indeterminate Powers
25:13
Example Problems for L'Hospital's Rule

38m 14s

Intro
0:00
Example I: Evaluate the Following Limit
0:17
Example II: Evaluate the Following Limit
2:45
Example III: Evaluate the Following Limit
6:54
Example IV: Evaluate the Following Limit
8:43
Example V: Evaluate the Following Limit
11:01
Example VI: Evaluate the Following Limit
14:48
Example VII: Evaluate the Following Limit
17:49
Example VIII: Evaluate the Following Limit
20:37
Example IX: Evaluate the Following Limit
25:16
Example X: Evaluate the Following Limit
32:44
Optimization Problems I

49m 59s

Intro
0:00
Example I: Find the Dimensions of the Box that Gives the Greatest Volume
1:23
Fundamentals of Optimization Problems
18:08
Fundamental #1
18:33
Fundamental #2
19:09
Fundamental #3
19:19
Fundamental #4
20:59
Fundamental #5
21:55
Fundamental #6
23:44
Example II: Demonstrate that of All Rectangles with a Given Perimeter, the One with the Largest Area is a Square
24:36
Example III: Find the Points on the Ellipse 9x² + y² = 9 Farthest Away from the Point (1,0)
35:13
Example IV: Find the Dimensions of the Rectangle of Largest Area that can be Inscribed in a Circle of Given Radius R
43:10
Optimization Problems II

55m 10s

Intro
0:00
Example I: Optimization Problem
0:13
Example II: Optimization Problem
17:34
Example III: Optimization Problem
35:06
Example IV: Revenue, Cost, and Profit
43:22
Newton's Method

30m 22s

Intro
0:00
Newton's Method
0:45
Newton's Method
0:46
Example I: Find x2 and x3
13:18
Example II: Use Newton's Method to Approximate
15:48
Example III: Find the Root of the Following Equation to 6 Decimal Places
19:57
Example IV: Use Newton's Method to Find the Coordinates of the Inflection Point
23:11
IV. Integrals
Antiderivatives

55m 26s

Intro
0:00
Antiderivatives
0:23
Definition of an Antiderivative
0:24
Antiderivative Theorem
7:58
Function & Antiderivative
12:10
x^n
12:30
1/x
13:00
e^x
13:08
cos x
13:18
sin x
14:01
sec² x
14:11
secxtanx
14:18
1/√(1-x²)
14:26
1/(1+x²)
14:36
-1/√(1-x²)
14:45
Example I: Find the Most General Antiderivative for the Following Functions
15:07
Function 1: f(x) = x³ -6x² + 11x - 9
15:42
Function 2: f(x) = 14√(x) - 27 4√x
19:12
Function 3: (fx) = cos x - 14 sinx
20:53
Function 4: f(x) = (x⁵+2√x )/( x^(4/3) )
22:10
Function 5: f(x) = (3e^x) - 2/(1+x²)
25:42
Example II: Given the Following, Find the Original Function f(x)
26:37
Function 1: f'(x) = 5x³ - 14x + 24, f(2) = 40
27:55
Function 2: f'(x) 3 sinx + sec²x, f(π/6) = 5
30:34
Function 3: f''(x) = 8x - cos x, f(1.5) = 12.7, f'(1.5) = 4.2
32:54
Function 4: f''(x) = 5/(√x), f(2) 15, f'(2) = 7
37:54
Example III: Falling Object
41:58
Problem 1: Find an Equation for the Height of the Ball after t Seconds
42:48
Problem 2: How Long Will It Take for the Ball to Strike the Ground?
48:30
Problem 3: What is the Velocity of the Ball as it Hits the Ground?
49:52
Problem 4: Initial Velocity of 6 m/s, How Long Does It Take to Reach the Ground?
50:46
The Area Under a Curve

51m 3s

Intro
0:00
The Area Under a Curve
0:13
Approximate Using Rectangles
0:14
Let's Do This Again, Using 4 Different Rectangles
9:40
Approximate with Rectangles
16:10
Left Endpoint
18:08
Right Endpoint
25:34
Left Endpoint vs. Right Endpoint
30:58
Number of Rectangles
34:08
True Area
37:36
True Area
37:37
Sigma Notation & Limits
43:32
When You Have to Explicitly Solve Something
47:56
Example Problems for Area Under a Curve

33m 7s

Intro
0:00
Example I: Using Left Endpoint & Right Endpoint to Approximate Area Under a Curve
0:10
Example II: Using 5 Rectangles, Approximate the Area Under the Curve
11:32
Example III: Find the True Area by Evaluating the Limit Expression
16:07
Example IV: Find the True Area by Evaluating the Limit Expression
24:52
The Definite Integral

43m 19s

Intro
0:00
The Definite Integral
0:08
Definition to Find the Area of a Curve
0:09
Definition of the Definite Integral
4:08
Symbol for Definite Integral
8:45
Regions Below the x-axis
15:18
Associating Definite Integral to a Function
19:38
Integrable Function
27:20
Evaluating the Definite Integral
29:26
Evaluating the Definite Integral
29:27
Properties of the Definite Integral
35:24
Properties of the Definite Integral
35:25
Example Problems for The Definite Integral

32m 14s

Intro
0:00
Example I: Approximate the Following Definite Integral Using Midpoints & Sub-intervals
0:11
Example II: Express the Following Limit as a Definite Integral
5:28
Example III: Evaluate the Following Definite Integral Using the Definition
6:28
Example IV: Evaluate the Following Integral Using the Definition
17:06
Example V: Evaluate the Following Definite Integral by Using Areas
25:41
Example VI: Definite Integral
30:36
The Fundamental Theorem of Calculus

24m 17s

Intro
0:00
The Fundamental Theorem of Calculus
0:17
Evaluating an Integral
0:18
Lim as x → ∞
12:19
Taking the Derivative
14:06
Differentiation & Integration are Inverse Processes
15:04
1st Fundamental Theorem of Calculus
20:08
1st Fundamental Theorem of Calculus
20:09
2nd Fundamental Theorem of Calculus
22:30
2nd Fundamental Theorem of Calculus
22:31
Example Problems for the Fundamental Theorem

25m 21s

Intro
0:00
Example I: Find the Derivative of the Following Function
0:17
Example II: Find the Derivative of the Following Function
1:40
Example III: Find the Derivative of the Following Function
2:32
Example IV: Find the Derivative of the Following Function
5:55
Example V: Evaluate the Following Integral
7:13
Example VI: Evaluate the Following Integral
9:46
Example VII: Evaluate the Following Integral
12:49
Example VIII: Evaluate the Following Integral
13:53
Example IX: Evaluate the Following Graph
15:24
Local Maxs and Mins for g(x)
15:25
Where Does g(x) Achieve Its Absolute Max on [0,8]
20:54
On What Intervals is g(x) Concave Up/Down?
22:20
Sketch a Graph of g(x)
24:34
More Example Problems, Including Net Change Applications

34m 22s

Intro
0:00
Example I: Evaluate the Following Indefinite Integral
0:10
Example II: Evaluate the Following Definite Integral
0:59
Example III: Evaluate the Following Integral
2:59
Example IV: Velocity Function
7:46
Part A: Net Displacement
7:47
Part B: Total Distance Travelled
13:15
Example V: Linear Density Function
20:56
Example VI: Acceleration Function
25:10
Part A: Velocity Function at Time t
25:11
Part B: Total Distance Travelled During the Time Interval
28:38
Solving Integrals by Substitution

27m 20s

Intro
0:00
Table of Integrals
0:35
Example I: Evaluate the Following Indefinite Integral
2:02
Example II: Evaluate the Following Indefinite Integral
7:27
Example IIII: Evaluate the Following Indefinite Integral
10:57
Example IV: Evaluate the Following Indefinite Integral
12:33
Example V: Evaluate the Following
14:28
Example VI: Evaluate the Following
16:00
Example VII: Evaluate the Following
19:01
Example VIII: Evaluate the Following
21:49
Example IX: Evaluate the Following
24:34
V. Applications of Integration
Areas Between Curves

34m 56s

Intro
0:00
Areas Between Two Curves: Function of x
0:08
Graph 1: Area Between f(x) & g(x)
0:09
Graph 2: Area Between f(x) & g(x)
4:07
Is It Possible to Write as a Single Integral?
8:20
Area Between the Curves on [a,b]
9:24
Absolute Value
10:32
Formula for Areas Between Two Curves: Top Function - Bottom Function
17:03
Areas Between Curves: Function of y
17:49
What if We are Given Functions of y?
17:50
Formula for Areas Between Two Curves: Right Function - Left Function
21:48
Finding a & b
22:32
Example Problems for Areas Between Curves

42m 55s

Intro
0:00
Instructions for the Example Problems
0:10
Example I: y = 7x - x² and y=x
0:37
Example II: x=y²-3, x=e^((1/2)y), y=-1, and y=2
6:25
Example III: y=(1/x), y=(1/x³), and x=4
12:25
Example IV: 15-2x² and y=x²-5
15:52
Example V: x=(1/8)y³ and x=6-y²
20:20
Example VI: y=cos x, y=sin(2x), [0,π/2]
24:34
Example VII: y=2x², y=10x², 7x+2y=10
29:51
Example VIII: Velocity vs. Time
33:23
Part A: At 2.187 Minutes, Which care is Further Ahead?
33:24
Part B: If We Shaded the Region between the Graphs from t=0 to t=2.187, What Would This Shaded Area Represent?
36:32
Part C: At 4 Minutes Which Car is Ahead?
37:11
Part D: At What Time Will the Cars be Side by Side?
37:50
Volumes I: Slices

34m 15s

Intro
0:00
Volumes I: Slices
0:18
Rotate the Graph of y=√x about the x-axis
0:19
How can I use Integration to Find the Volume?
3:16
Slice the Solid Like a Loaf of Bread
5:06
Volumes Definition
8:56
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
12:18
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
19:05
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
25:28
Volumes II: Volumes by Washers

51m 43s

Intro
0:00
Volumes II: Volumes by Washers
0:11
Rotating Region Bounded by y=x³ & y=x around the x-axis
0:12
Equation for Volumes by Washer
11:14
Process for Solving Volumes by Washer
13:40
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
15:58
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
25:07
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
34:20
Example IV: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
44:05
Volumes III: Solids That Are Not Solids-of-Revolution

49m 36s

Intro
0:00
Solids That Are Not Solids-of-Revolution
0:11
Cross-Section Area Review
0:12
Cross-Sections That Are Not Solids-of-Revolution
7:36
Example I: Find the Volume of a Pyramid Whose Base is a Square of Side-length S, and Whose Height is H
10:54
Example II: Find the Volume of a Solid Whose Cross-sectional Areas Perpendicular to the Base are Equilateral Triangles
20:39
Example III: Find the Volume of a Pyramid Whose Base is an Equilateral Triangle of Side-Length A, and Whose Height is H
29:27
Example IV: Find the Volume of a Solid Whose Base is Given by the Equation 16x² + 4y² = 64
36:47
Example V: Find the Volume of a Solid Whose Base is the Region Bounded by the Functions y=3-x² and the x-axis
46:13
Volumes IV: Volumes By Cylindrical Shells

50m 2s

Intro
0:00
Volumes by Cylindrical Shells
0:11
Find the Volume of the Following Region
0:12
Volumes by Cylindrical Shells: Integrating Along x
14:12
Volumes by Cylindrical Shells: Integrating Along y
14:40
Volumes by Cylindrical Shells Formulas
16:22
Example I: Using the Method of Cylindrical Shells, Find the Volume of the Solid
18:33
Example II: Using the Method of Cylindrical Shells, Find the Volume of the Solid
25:57
Example III: Using the Method of Cylindrical Shells, Find the Volume of the Solid
31:38
Example IV: Using the Method of Cylindrical Shells, Find the Volume of the Solid
38:44
Example V: Using the Method of Cylindrical Shells, Find the Volume of the Solid
44:03
The Average Value of a Function

32m 13s

Intro
0:00
The Average Value of a Function
0:07
Average Value of f(x)
0:08
What if The Domain of f(x) is Not Finite?
2:23
Let's Calculate Average Value for f(x) = x² [2,5]
4:46
Mean Value Theorem for Integrate
9:25
Example I: Find the Average Value of the Given Function Over the Given Interval
14:06
Example II: Find the Average Value of the Given Function Over the Given Interval
18:25
Example III: Find the Number A Such that the Average Value of the Function f(x) = -4x² + 8x + 4 Equals 2 Over the Interval [-1,A]
24:04
Example IV: Find the Average Density of a Rod
27:47
VI. Techniques of Integration
Integration by Parts

50m 32s

Intro
0:00
Integration by Parts
0:08
The Product Rule for Differentiation
0:09
Integrating Both Sides Retains the Equality
0:52
Differential Notation
2:24
Example I: ∫ x cos x dx
5:41
Example II: ∫ x² sin(2x)dx
12:01
Example III: ∫ (e^x) cos x dx
18:19
Example IV: ∫ (sin^-1) (x) dx
23:42
Example V: ∫₁⁵ (lnx)² dx
28:25
Summary
32:31
Tabular Integration
35:08
Case 1
35:52
Example: ∫x³sinx dx
36:39
Case 2
40:28
Example: ∫e^(2x) sin 3x
41:14
Trigonometric Integrals I

24m 50s

Intro
0:00
Example I: ∫ sin³ (x) dx
1:36
Example II: ∫ cos⁵(x)sin²(x)dx
4:36
Example III: ∫ sin⁴(x)dx
9:23
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sin^m) (x) (cos^p) (x) dx
15:59
#1: Power of sin is Odd
16:00
#2: Power of cos is Odd
16:41
#3: Powers of Both sin and cos are Odd
16:55
#4: Powers of Both sin and cos are Even
17:10
Example IV: ∫ tan⁴ (x) sec⁴ (x) dx
17:34
Example V: ∫ sec⁹(x) tan³(x) dx
20:55
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sec^m) (x) (tan^p) (x) dx
23:31
#1: Power of sec is Odd
23:32
#2: Power of tan is Odd
24:04
#3: Powers of sec is Odd and/or Power of tan is Even
24:18
Trigonometric Integrals II

22m 12s

Intro
0:00
Trigonometric Integrals II
0:09
Recall: ∫tanx dx
0:10
Let's Find ∫secx dx
3:23
Example I: ∫ tan⁵ (x) dx
6:23
Example II: ∫ sec⁵ (x) dx
11:41
Summary: How to Deal with Integrals of Different Types
19:04
Identities to Deal with Integrals of Different Types
19:05
Example III: ∫cos(5x)sin(9x)dx
19:57
More Example Problems for Trigonometric Integrals

17m 22s

Intro
0:00
Example I: ∫sin²(x)cos⁷(x)dx
0:14
Example II: ∫x sin²(x) dx
3:56
Example III: ∫csc⁴ (x/5)dx
8:39
Example IV: ∫( (1-tan²x)/(sec²x) ) dx
11:17
Example V: ∫ 1 / (sinx-1) dx
13:19
Integration by Partial Fractions I

55m 12s

Intro
0:00
Integration by Partial Fractions I
0:11
Recall the Idea of Finding a Common Denominator
0:12
Decomposing a Rational Function to Its Partial Fractions
4:10
2 Types of Rational Function: Improper & Proper
5:16
Improper Rational Function
7:26
Improper Rational Function
7:27
Proper Rational Function
11:16
Proper Rational Function & Partial Fractions
11:17
Linear Factors
14:04
Irreducible Quadratic Factors
15:02
Case 1: G(x) is a Product of Distinct Linear Factors
17:10
Example I: Integration by Partial Fractions
20:33
Case 2: D(x) is a Product of Linear Factors
40:58
Example II: Integration by Partial Fractions
44:41
Integration by Partial Fractions II

42m 57s

Intro
0:00
Case 3: D(x) Contains Irreducible Factors
0:09
Example I: Integration by Partial Fractions
5:19
Example II: Integration by Partial Fractions
16:22
Case 4: D(x) has Repeated Irreducible Quadratic Factors
27:30
Example III: Integration by Partial Fractions
30:19
VII. Differential Equations
Introduction to Differential Equations

46m 37s

Intro
0:00
Introduction to Differential Equations
0:09
Overview
0:10
Differential Equations Involving Derivatives of y(x)
2:08
Differential Equations Involving Derivatives of y(x) and Function of y(x)
3:23
Equations for an Unknown Number
6:28
What are These Differential Equations Saying?
10:30
Verifying that a Function is a Solution of the Differential Equation
13:00
Verifying that a Function is a Solution of the Differential Equation
13:01
Verify that y(x) = 4e^x + 3x² + 6x + e^π is a Solution of this Differential Equation
17:20
General Solution
22:00
Particular Solution
24:36
Initial Value Problem
27:42
Example I: Verify that a Family of Functions is a Solution of the Differential Equation
32:24
Example II: For What Values of K Does the Function Satisfy the Differential Equation
36:07
Example III: Verify the Solution and Solve the Initial Value Problem
39:47
Separation of Variables

28m 8s

Intro
0:00
Separation of Variables
0:28
Separation of Variables
0:29
Example I: Solve the Following g Initial Value Problem
8:29
Example II: Solve the Following g Initial Value Problem
13:46
Example III: Find an Equation of the Curve
18:48
Population Growth: The Standard & Logistic Equations

51m 7s

Intro
0:00
Standard Growth Model
0:30
Definition of the Standard/Natural Growth Model
0:31
Initial Conditions
8:00
The General Solution
9:16
Example I: Standard Growth Model
10:45
Logistic Growth Model
18:33
Logistic Growth Model
18:34
Solving the Initial Value Problem
25:21
What Happens When t → ∞
36:42
Example II: Solve the Following g Initial Value Problem
41:50
Relative Growth Rate
46:56
Relative Growth Rate
46:57
Relative Growth Rate Version for the Standard model
49:04
Slope Fields

24m 37s

Intro
0:00
Slope Fields
0:35
Slope Fields
0:36
Graphing the Slope Fields, Part 1
11:12
Graphing the Slope Fields, Part 2
15:37
Graphing the Slope Fields, Part 3
17:25
Steps to Solving Slope Field Problems
20:24
Example I: Draw or Generate the Slope Field of the Differential Equation y'=x cos y
22:38
VIII. AP Practic Exam
AP Practice Exam: Section 1, Part A No Calculator

45m 29s

Intro
0:00
Exam Link
0:10
Problem #1
1:26
Problem #2
2:52
Problem #3
4:42
Problem #4
7:03
Problem #5
10:01
Problem #6
13:49
Problem #7
15:16
Problem #8
19:06
Problem #9
23:10
Problem #10
28:10
Problem #11
31:30
Problem #12
33:53
Problem #13
37:45
Problem #14
41:17
AP Practice Exam: Section 1, Part A No Calculator, cont.

41m 55s

Intro
0:00
Problem #15
0:22
Problem #16
3:10
Problem #17
5:30
Problem #18
8:03
Problem #19
9:53
Problem #20
14:51
Problem #21
17:30
Problem #22
22:12
Problem #23
25:48
Problem #24
29:57
Problem #25
33:35
Problem #26
35:57
Problem #27
37:57
Problem #28
40:04
AP Practice Exam: Section I, Part B Calculator Allowed

58m 47s

Intro
0:00
Problem #1
1:22
Problem #2
4:55
Problem #3
10:49
Problem #4
13:05
Problem #5
14:54
Problem #6
17:25
Problem #7
18:39
Problem #8
20:27
Problem #9
26:48
Problem #10
28:23
Problem #11
34:03
Problem #12
36:25
Problem #13
39:52
Problem #14
43:12
Problem #15
47:18
Problem #16
50:41
Problem #17
56:38
AP Practice Exam: Section II, Part A Calculator Allowed

25m 40s

Intro
0:00
Problem #1: Part A
1:14
Problem #1: Part B
4:46
Problem #1: Part C
8:00
Problem #2: Part A
12:24
Problem #2: Part B
16:51
Problem #2: Part C
17:17
Problem #3: Part A
18:16
Problem #3: Part B
19:54
Problem #3: Part C
21:44
Problem #3: Part D
22:57
AP Practice Exam: Section II, Part B No Calculator

31m 20s

Intro
0:00
Problem #4: Part A
1:35
Problem #4: Part B
5:54
Problem #4: Part C
8:50
Problem #4: Part D
9:40
Problem #5: Part A
11:26
Problem #5: Part B
13:11
Problem #5: Part C
15:07
Problem #5: Part D
19:57
Problem #6: Part A
22:01
Problem #6: Part B
25:34
Problem #6: Part C
28:54
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Lecture Comments (8)

1 answer

Last reply by: Professor Hovasapian
Fri Apr 7, 2017 6:40 PM

Post by Francisco Gonzalez on April 3, 2017

Professor Hovasapian, are you doing the Recording of the lessons for BC anytime soon as you have stated: "Recording of the lessons for BC should be finished in the next 2-3 months"

I wonder why Educator is not uploading new lessons, I asked but they didn't really say why they stopped, just wondering.

1 answer

Last reply by: Mohammed Jaweed
Fri Mar 18, 2016 12:08 AM

Post by Mohammed Jaweed on March 18, 2016

The integral of cos(x) is not -sin(x). It is sin(x).

3 answers

Last reply by: Professor Hovasapian
Thu Jan 7, 2016 11:01 PM

Post by Gautham Padmakumar on January 2, 2016

at 2:24 for the differential notation, why do we write it as du = f'(x)
why not du = f'(x) dx

thanks! helpful videos as always. also curious as to when the Calculus BC course by you will be uploaded? :)

Integration by Parts

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Integration by Parts 0:08
    • The Product Rule for Differentiation
    • Integrating Both Sides Retains the Equality
    • Differential Notation
  • Example I: ∫ x cos x dx 5:41
  • Example II: ∫ x² sin(2x)dx 12:01
  • Example III: ∫ (e^x) cos x dx 18:19
  • Example IV: ∫ (sin^-1) (x) dx 23:42
  • Example V: ∫₁⁵ (lnx)² dx 28:25
  • Summary 32:31
  • Tabular Integration 35:08
    • Case 1
    • Example: ∫x³sinx dx
    • Case 2
    • Example: ∫e^(2x) sin 3x

Transcription: Integration by Parts

Hello, welcome back to www.educator.com, and welcome back to AP Calculus.0000

Today, we are going to talk about the technique called integration by parts, let us jump right on in.0004

Let us recall the product rule from differential calculus.0011

We have the product rule for differentiation says f × g’ is equal to f’ g + fg’.0019

If I integrate both sides, I can retain the equality.0049

Integrating both sides retains the equality, as we know.0058

Anything we do to one side, as long as we do it to the other, everything is fine.0067

We have got the integral of fg’ = the integral of f’ g + fg’.0071

The integral of the derivative, this goes away.0089

That just becomes fg is equal to, and the integral operator is linear.0093

I can essentially distribute it out, it becomes the integral of f’ g + the integral of fg’.0097

Now I’m going to rearrange to get the following.0107

I get the integral of fg’ is equal to fg - the integral of f’ g.0114

This is our formula for something called integration by parts.0129

Again, do not worry we are going to actually explain this, much of this actually makes sense0134

once you actually see some examples.0138

In differential notation, it is going to look like this.0141

That is how we are actually going to be using it.0144

In differential notation, u is our function of x.0147

I’m going to arrange it this way.0171

u is going to be our f(x), it will make sense why I’m arranging this.0172

Our v is going to equal our g(x).0178

Our dv is equal to g’(x).0185

Never mind, let us stick to what we have got.0194

u is our f(x), v is our g(x), du is our f’(x), and dv is our g’(x).0199

Based on what we just wrote in the previous thing.0213

With that notation, what we have is the following.0218

We have the integral of u dv f(x) g’ fg’ is equal to uv fg - the integral of v du gf’.0222

This is the way we are actually going to use the integration by parts.0244

If we have an integrand, if we are faced with an integrand which is a product of two functions0249

and we can express u dv, in other words, this product,0281

if we can express u dv as some function 1 × some function 2,0294

then, we can solve the integral by using the right side of the equality.0305

Again, examples will make this clear.0331

Do not worry if you do not understand anything that I have just wrote symbolically.0333

Examples are going to make this immediately clear.0337

Let us jump right on in to our first example.0341

This is going to be the integral of x cos x.0344

It looks like we are missing a little bit of our top integral sign there.0347

We want to transform this to an easier integral.0352

We want to use integration by parts.0356

This is the integral of a function × another function.0358

One function is x, one function is cos x.0362

We want to use integration by parts to express as a simpler expression.0365

Something that we actually can integrate directly.0390

Here is how we are going to do it.0407

Let me go ahead and go to red here.0410

We have the integral of u dv = uv - the integral of v du.0412

Write the formula down and now we are going to use the formula.0423

Here I’m going to let u is equal to x dv.0427

This is our u dv, this is what we are doing.0436

We are looking at it as a u dv.0438

This is my u and this cos x, I’m going to call this my dv.0441

dv = cos(x).0447

What is du?0453

du is just 1, or 1 dx.0455

It is okay, we are using differentials.0459

What is v?0460

If dv = cos x, integrating this, what is the integral of cos x?0461

It is – sin x.0468

Here we are differentiating.0475

All I have is I'm looking at the integrand as some function, as some u and as some dv.0476

The u, I differentiate down.0487

The dv, I integrate up.0489

Now I use this side.0491

What I end up getting is the integral of x cos x is equal to this side.0496

They are the same thing, I can substitute uv.0507

Here is my u and here is my v, that is equal to -x × sin(x) - the integral of v × du.0510

v is -sin x × du.0523

du is just 1, that is easy.0529

This is – x sin x + the integral of sin x dx.0534

I get – x sin x, the integral of sin is,0544

I’m sorry, this is wrong.0552

The integral of cos is just sin.0553

When we integrated the cos x, the integral of cos x is sin x.0567

This is positive, this is positive, and this is positive.0573

We have used this side and we have actually created a simpler integral, based on our choice of u and dv.0581

The integral of sin x is just -cos x.0593

This becomes + cos x and +c.0596

With differentiation, the product rule is really simple.0604

It is just first × the derivative of the other + that one × the derivative of the first.0607

With integration, when you have a product of two functions, we do not have a complete closed form expression.0612

The way we do for differentiation.0622

But we can arrange it in such a way, such that the interval that we get on the right side0624

is a little bit easier than the original integral.0630

Something that we actually can solve, based on our choice of u and dv.0633

It is possible that the order is different.0638

Maybe this could have been the u and this could have been the dv.0641

Sometimes you are going to pick the wrong order and you realize that the integral you get is not any easier than the original.0644

You have to go back and switch the order.0649

Pick something else for u and pick something else for dv.0652

We will see some examples of that.0655

By integration by parts, by using integration by parts,0660

the original integral was equal to an expression to the right side,0690

where the integral was easier to handle.0712

That is it, that is really all we have done.0715

Let us try another example, this one is going to be the integral of x² sin 2x.0721

It seem to be missing the upper fourth of our integral signs.0729

In this case, let us go ahead and again write the expression.0733

Let me go ahead and do this in red.0737

We have the integral of u dv is equal to uv - the integral of v du.0740

Let us just go ahead and rewrite the formula over and over again, until it sticks in your mind.0747

We have to make a choice, this is a function x² × sin(2x).0752

Which one are we going to take for u, which one are we going to take for dv?0756

Pick one and hopefully you will get lucky.0762

I'm going to take u is equal to x² and I'm going to take my dv equal to sin(2x).0764

du is equal to 2x, and v, I’m going to integrate up.0774

The integral of sin 2x is equal to - ½ cos 2x.0781

I really hope to God that I’m getting these integrals correct.0789

Now we have got, let us use our formula.0792

The integral of u dv is equal to uv - the integral v du.0795

The integral of x² × sin (2x) dx is equal to uv which is x² × -1/2 cos 2x - the integral of v du.0801

v is -1/2 cos 2x × 2x dx.0820

Let us rewrite this, that = -1/2 x² cos(2x) +, the negative and negative becomes +,0841

the 2 and 2 cancel, + the integral of x × cos(2x) dx.0856

It does not necessarily look any easier, does it?0872

Because now all of a sudden we have this part which is nice,0875

but now we have another integral which is a product of a function and another function.0878

We are going to do another cycle of this.0884

Now we will just take this, we do a second round of integration by parts.0886

This time I’m going to set u equal to x.0905

I’m going to set dv equal to cos 2x.0911

The derivative of u = 1, and v, the integral of cos 2x is ½ sin 2x.0918

Therefore, this integral becomes just this integral.0930

It becomes uv - the integral of v du.0937

uv is going to be ½ x × sin(2x) - the integral of v du which is ½ sin 2x × du 1 dx.0943

Now let us take a look at this integral right here.0978

This is nice, this is just the integral of ½ sin 2x dx.0985

This integral, we can handle, now we can stop.0992

We take our first term.0995

This integral right here, the original integral is equal to this term which is -1/2 x² cos 2x + this which is this.0996

I will take this term + ½ x sin 2x.1015

I will go ahead and I will actually integrate and that.1026

The integral of ½ sin 2x is going to be + 1/4 cos(2x) + c.1030

That is my final answer from my original integral.1051

My original integral was a product.1055

I used u dv, I used the integration by parts formula to find this first expression right here.1057

This one was another integral where you had a product function × a function.1065

I did a second round of integration by parts and I have got this expression.1071

This time around, the integral in this expression, for the right side of the integration by parts formula,1075

happen to be a simple integral that we can solve.1081

We took this term and this term, and then we find the integral and we ended up with our final answer.1083

There you go, integration by parts.1090

Sometimes, integration by parts is going to require multiple rounds.1092

Let us do e ⁺x cos x dx.1100

Let us see what we have got.1106

Once again, let us write our integral formula.1108

The integral of u dv is equal to uv - the integral of v du.1114

I'm going to take my u equal to e ⁺x.1123

I’ m going to take my dv equal to cos x.1130

My du is going to equal e ⁺x.1137

My v is going to equal sin x.1139

Therefore, the integral of u dv is equal to uv – v du.1145

You know what, I will just write out the whole thing.1155

The integral of e ⁺x cos x dx = uv e ⁺x × sin x - the integral of v du e ⁺x sin x dx.1156

Again, we have a function of two.1179

We are going to do a second round here.1182

Once again, we are going to do a second round.1184

Let me do this second round in blue.1190

This time u is equal to e ⁺x dv is equal to sin x du is equal to e ⁺x v.1196

The integral of sin x is -cos x.1207

Therefore, this integral is going to yield, it is going to be uv - the integral v du.1211

uv – e ⁺x cos x - the integral of v du – v du e ⁺x cos x.1219

- and -, I will turn this into a +.1237

It is going to be e ⁺x cos x dx.1239

That is interesting.1245

What have I got?1250

Now I have got this thing which is a solution of this thing.1257

Let me bring back my first term.1263

I have the integral, the original integral of e ⁺x cos x dx is equal to this first term e ⁺x sin x - this integral1265

which is this thing – e ⁺x cos x + the integral of e ⁺x cos x dx.1287

Now I have this right here.1304

Negative and negative becomes +.1310

This negative distributes over here that becomes -.1312

We end up with the integral of e ⁺x cos x dx which is our original.1315

I get e ⁺x sin x + e ⁺x cos x - the integral of e ⁺x cos x dx.1324

Notice that and that are the same.1339

I’m going to move this over to that side because this is an equality.1351

I can move it over to that side.1355

I’m going to get 2 × the integral of e ⁺x cos x dx is equal to e ⁺x sin x + e ⁺x cos x, then I divide by 2.1357

The integral of e ⁺x cos x dx, I’m going to get a closed form expression.1372

e ⁺x sin x + e ⁺x cos x/ 2 + c.1380

There you go, I hope that made sense.1389

This time, what we did in the second round ended up giving me something that actually,1393

the integral was a repeat of the original integral.1399

Because it is an equality, I can just bring whatever term is here over to the other side1403

and divide by whatever constant is in front to leave me an integral, and then on the right side, a closed form expression.1407

I hope that made sense.1415

Let us see what we have got here.1420

Once again, let us go ahead and finish our integral.1425

The integral of inv sin x dx.1428

In this case, you notice that it is not a product, it is just one function, inv sin x dx,1432

but we can use integration by parts.1437

The integral, because there is no elementary integral for that.1442

Let us see what we have got.1445

The integral of u dv is equal to u × v - the integral of v du.1446

We write our formula.1455

There is a 1 here, let us try u is equal to 1.1458

dv = integral sin x, I did not really do anything because I can integrate this because that is what I’m trying to integrate.1466

That order is not going to work.1478

Let us try u is equal to inv sin(x) and dv is equal to 1.1497

This we can do, the du, we know that the derivative of the inv sin that is just 1/√1 - x².1506

v is equal to the integral of 1 is just x.1515

Therefore, the integral of inv sin(x) dx is equal to uv.1521

u × v is equal to x × the inv sin(x) - the integral of v du x/ 1 - x², under the radical dx.1530

This one, we can actually probably solve with just the u substitution.1555

Let me go to red.1560

I’m going to let u equal to 1 - x², du = -2x dx.1561

I have got x dx equal to du/-2.1574

Therefore, this integral is actually equal to, the integral of -du/2 × u⁻¹/21582

= -1/2 the integral of u⁻¹/2 du, which is equal to -1/2 u ^½ / ½.1602

That cancels, I'm left with just u ^-½ which is equal to -√1 - x².1625

This integral is equal to that.1640

Therefore, our answer is the integral of inv sin(x) dx is equal to1645

x × inv sin(x) - this integral which is -√1 - x².1660

It is equal to x × inv sin(x) + √1 - x² + c.1672

Now we have an integral for the inverse sin using integration by parts.1685

Again, just because you have one function here, you can still use the technique of integration by parts1691

by choosing one of them to equal 1, usually that dv to equal 1 and seeing where you go.1698

Let us see what is next here.1707

The integral from 1 to 5, this is a definite integral of ln x².1710

Let me see.1717

Let us go back to black here, let us rewrite our formula.1722

The integral of u dv is equal to u × v - the integral of v du.1725

Again, I'm going to take this time on u is equal to ln(x)².1735

I’m going to take dv equal to 1.1743

du is equal to 2 ln x/ x.1749

v is equal to x.1758

It is kind of convenient.1763

Therefore, the integral from 1 to 5 of ln x² dx is equal to this, uv, u × v.1765

It is equal to x × ln(x)² - the integral of v du.1780

v du x × 2 ln x/ x dx.1797

That worked out really beautifully, = x × ln x² - 2 × the integral of ln x dx.1806

Let us see, might have to do a second round here.1824

Let us go to blue. We have got ln x, what is the integral of ln x?1834

Let us take u equal to ln x.1838

Let us take dv equal to 1, du = 1/x.1843

v is equal to x.1851

Therefore this is going to give us uv, u × v.1854

This is going to be x ln x - the integral of v du.1859

x × 1/x dx, this is equal to x ln x.1867

x/x cancels, the integral of 1 is just x.1875

Therefore, our final answer is equal to that x ln x² from 1 to 5,1880

this is a definite integral, -2 × this integral which is this thing x ln x - x from 1 to 5.1906

That is our answer.1923

Of course, you have to evaluate this upper and lower thing.1926

I will leave that to you.1932

But that is it, first application, second application, final answer.1933

And then, go ahead and work it out.1938

Let us see what is next.1944

The idea is always the same.1951

The idea is always the same.1957

Mainly, given the integral of f × g, choose an appropriate u and dv.1965

Either f is going to be u and g is going to be dv or g is going to be u and f is going to be dv,1988

whichever gives you the simplest integral on the right,1996

that you can either deal with directly or do another round, dv, and see where things go.1999

If you hit a wall, in other words, if you end up with an integral on the right which is intractable by any technique,2028

just switch original choice.2040

Just switch your choice of u and dv, and see where that one goes.2052

Again, the integral that you get on the right is either going to be directly integrable2060

or you would have to go through a second round.2065

Or you might have to use another technique, whatever it is,2069

u substitution or trigonometric substitution, whatever it is that you need.2072

That is pretty much what we are doing.2078

Now I want to introduce something called tabular integration.2081

From what it looks like, this is definitely not something that is actually taught anymore.2086

I always taught it when I learn calculus and I think it is a really nice powerful technique2092

to make your life a lot easier and a lot faster.2096

I will leave it up to you and I will leave it up to your teacher to decide2099

whether this is something that they will accept on an exam or something like that.2102

Let us talk about tabular integration.2106

There is a technique, this technique, greatly reduces the work involved in working with integration by parts.2121

We will do it with cases.2153

Case 1, when one of the functions that you choose, when one of the functions differentiates down to 0,2155

one of the functions under the integrand differentiates down to 0, and the other integrates up infinitely.2175

This is the first case, an example.2197

Now if you are faced with the integral, the integral of x³ sin(x) dx.2204

Let us go to red here.2211

This x³ will eventually differentiate down to 0.2213

x³, 3x², 6x, 6, 0.2216

Sin x, it integrates up infinitely.2222

Sin x - cos x -, it goes on and on and on.2226

Here is how we actually do this.2231

We let u equal to, this is really great.2234

Here is our u and here is our dv.2242

u is equal to x³, dv is equal to sin x, differentiate this straight all the way down to 0.2246

The first one gives you 3x².2254

The second gives you 6x.2258

The third gives you 6, the last gives you 0, integrate up.2261

1, 2, 3, 4, 1 differentiation, 2 differentiation, 3 differentiation, 4 differentiation, so integrate.2268

The integral of sin x is -cos x, the integral of -cos x is – sin x.2275

The integral of –sin x is cos x and the integral of cos x is sin x.2284

Now you put them together like this.2292

With a + sign, with a – sign, alternating.2298

With a + sign, with a – sign.2304

This, when you do integration by parts, you are going to end up doing 1, 2, 3 rounds, 3, 4 rounds.2309

If you use tabular integration, your final answer is going to be this.2317

It is going to equal x³ – cos.2322

You are going to put these two together.2325

It is going to be - x³ cos x.2326

Let me do blue, you are going to put this and this together with a + sign.2332

It is going to be, you are going to put this and this together with the – sign.2338

- and - becomes + 3x² sin x.2342

This and this, +6x cos x.2348

And then, this and this, -6 sin x + c.2355

That is it, look at how much time I have saved.2365

I probably saved about 10 minutes of work.2367

Tabular integration, when you have one function, when one of the function is under the integrand2369

actually differentiates down to 0, the other one integrates infinitely, go ahead and do that.2374

Differentiate it down to 0, that many times, integrate that many times up infinitely.2380

And then, connect the original with the next one down, the original with the next one down,2385

the original with the next one down, the original until you reach the end of this one.2390

And then alternate signs, + -, +-, that gives you what you would have gotten,2395

if you had gone through four rounds of integration by parts.2399

This is tabular integration and it works really beautifully.2405

The first case is when one of the functions differentiates down to 0 and the other integrates infinitely.2408

Case 2, I actually could have presented this as one case, which I probably should have, now that I'm looking at it.2416

But that is okay, I will do it like this.2422

Case 2, this is when both functions,2428

I better write it out explicitly.2442

When one function differentiates infinitely and the other integrates infinitely.2447

Here is an example, example that we are going to be looking at is the integral of e ⁺2x sin 3x.2474

Here is how we deal with this one.2491

Let us go ahead and go to red.2493

I'm going to let u, I’m going to have dv.2495

u, I’m going to call e ⁺2x u, it is my choice for u.2500

sin(3x) is going to be my dv.2505

You are going to differentiate, they differentiated and integrate infinitely.2511

The question is where do you stop?2514

Before, you can stop at 0, you cannot go any further.2515

Here is how you are going to do it.2518

Differentiate, this is going to be e ⁺2x.2520

Integrate, this is going to be -1/3 cos 3x.2524

Differentiate, it is time to go back and fourth.2535

This is 2e ⁺2x, my apologies.2538

When we differentiate again, this is going to be 4e ⁺2x.2542

When we integrate again, we are going to get -1/9 sin(3x).2546

Here is where you stop.2559

Stop when you have integrated dv to a point where you have a version of the original dv.2563

Our original dv was sin 3x.2599

I integrate, I integrate, I have sin 3x again.2603

It is -1/9 sin 3x but it is still sin 3x.2606

It is a version of the original, that is when you stop.2609

What do you put together, here is what you put together.2613

You do the same thing, except one little extra of this + alternating this - the last one is this + with an integral sign.2616

Here is what it looks like.2636

It is going to be -1/3 e ⁺2x cos 3x + 2/9, e ⁺2x sin 3x.2638

Put these two together under an integral sign.2669

It is going to be, I will just write this as + the integral - 4/9 e ⁺2x sin 3x.2673

Notice what you have got, you have e ⁺2x sin 3x 4/9.2691

The original was e ⁺2x sin 3x.2697

You can do that where you move it over to the other side and you divide.2700

What you get here is the following.2704

We have the original e ⁺2x sin 3x and we set that is equal to -1/3 e ⁺2x cos(3x) + 2/9 e ⁺x sin 3x - 4/9.2710

The integral of e ⁺x sin 3x dx.2739

This and this, because this is an original version of that, you can move it over here.2746

What you are going to end up with is 4/9 + 9/9, you are going to end up with 13/9 ×2754

the integral of e ⁺2x sin 3x dx is equal to -1/3 e ⁺2x cos 3x + 2/9 e ⁺x sin 3x.2773

Of course, you divide by 13/9, and then you get your final answer.2793

Your final answer is going to be the integral of e ⁺2x sin 3x dx is going to equal,2798

After you divide by that, you are going to end up with -3/13 e ⁺2x cos 3x + 2/13 e ⁺x sin 3x + c.2810

There you go.2828

Now case 1 is just a special case of case 2, here is why.2831

Remember with case 1, what we had was this.2850

We had u and we had dv, and we had x³, and we had sin(x).2854

This goes to 3x², this goes to 6x, this goes to 6, and this goes to 0.2861

The sin x goes to -cos x, -cos x goes to -sin x.2871

-sin x integrates to cos x, and cos x integrates to sin x.2879

We said, what you are going to take is the following.2886

You are going to take this and that, with a + sign.2889

This and that, with a – sign.2894

This and that, with a + sign.2896

This and that, with a – sign.2899

The final one, it is going to be this and that, with a + sign under the integral sign.2901

Because case 2, it was always the last two.2909

As long as you have reached some version of the original dv,2914

that last thing that you get is multiplied by this and it falls under the integral sign.2919

If I wrote this out, I would get the following.2927

I would get x³ - x³ cos x, that takes care of that and that.2929

I would get + 3x² sin x, that takes care of that and that.2938

I would get + 6x cos x, it takes care of those two.2946

I would get -6 sin x, that takes care of those two.2952

I would get + the integral of 0 × sin x dx.2957

The integral of 0 does not really matter.2963

It is 0, it is a constant, when you integrate that.2971

There is your constant.2975

This case 1 is really just a special case of case 2.2977

Differentiate down to 0, integrate up.2984

Put them together, alternating signs.2987

Differentiate, integrate up until you end up getting a version of the original dv.2992

The last one, you put together under the integral sign.2998

This is tabular integration, I use it all the time.3002

Case 1 is the one you actually use most often.3007

Case 2, I do not actually use case 2 all that often.3009

I tend to use it more and more now.3015

This whole last part, I do not know, some people are a little weary of that but it actually does work.3018

I would not worry about it all that much.3024

In any case, I hope that helps.3026

Take good care and thank you for joining us here at www.educator.com.3028

We will see you next time, bye.3031

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