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Professor Murray

Trigonometric Substitutions

Slide Duration:Table of Contents

I. Advanced Integration Techniques

Integration by Parts

24m 52s

- Intro0:00
- Important Equation0:07
- Where It Comes From (Product Rule)0:20
- Why Use It?0:35
- Lecture Example 11:24
- Lecture Example 23:30
- Shortcut: Tabular Integration7:34
- Example7:52
- Lecture Example 310:00
- Mnemonic: LIATE14:44
- Ln, Inverse, Algebra, Trigonometry, e15:38
- Additional Example 4-1
- Additional Example 5-2

Integration of Trigonometric Functions

25m 30s

- Intro0:00
- Important Equation0:07
- Powers (Odd and Even)0:19
- What To Do1:03
- Lecture Example 11:37
- Lecture Example 23:12
- Half-Angle Formulas6:16
- Both Powers Even6:31
- Lecture Example 37:06
- Lecture Example 410:59
- Additional Example 5-1
- Additional Example 6-2

Trigonometric Substitutions

30m 9s

- Intro0:00
- Important Equations0:06
- How They Work0:35
- Example1:45
- Remember: du and dx2:50
- Lecture Example 13:43
- Lecture Example 210:01
- Lecture Example 312:04
- Additional Example 4-1
- Additional Example 5-2

Partial Fractions

41m 22s

- Intro0:00
- Overview0:07
- Why Use It?0:18
- Lecture Example 11:21
- Lecture Example 26:52
- Lecture Example 313:28
- Additional Example 4-1
- Additional Example 5-2

Integration Tables

20m

- Intro0:00
- Using Tables0:09
- Match Exactly0:32
- Lecture Example 11:16
- Lecture Example 25:28
- Lecture Example 38:51
- Additional Example 4-1
- Additional Example 5-2

Trapezoidal Rule, Midpoint Rule, Left/Right Endpoint Rule

22m 36s

- Intro0:00
- Trapezoidal Rule0:13
- Graphical Representation0:20
- How They Work1:08
- Formula1:47
- Why a Trapezoid?2:53
- Lecture Example 15:10
- Midpoint Rule8:23
- Why Midpoints?8:56
- Formula9:37
- Lecture Example 211:22
- Left/Right Endpoint Rule13:54
- Left Endpoint14:08
- Right Endpoint14:39
- Lecture Example 315:32
- Additional Example 4-1
- Additional Example 5-2

Simpson's Rule

21m 8s

- Intro0:00
- Important Equation0:03
- Estimating Area0:28
- Difference from Previous Methods0:50
- General Principle1:09
- Lecture Example 13:49
- Lecture Example 26:32
- Lecture Example 39:07
- Additional Example 4-1
- Additional Example 5-2

Improper Integration

44m 18s

- Intro0:00
- Horizontal and Vertical Asymptotes0:04
- Example: Horizontal0:16
- Formal Notation0:37
- Example: Vertical1:58
- Formal Notation2:29
- Lecture Example 15:01
- Lecture Example 27:41
- Lecture Example 311:32
- Lecture Example 415:49
- Formulas to Remember18:26
- Improper Integrals18:36
- Lecture Example 521:34
- Lecture Example 6 (Hidden Discontinuities)26:51
- Additional Example 7-1
- Additional Example 8-2

II. Applications of Integrals, part 2

Arclength

23m 20s

- Intro0:00
- Important Equation0:04
- Why It Works0:49
- Common Mistake1:21
- Lecture Example 12:14
- Lecture Example 26:26
- Lecture Example 310:49
- Additional Example 4-1
- Additional Example 5-2

Surface Area of Revolution

28m 53s

- Intro0:00
- Important Equation0:05
- Surface Area0:38
- Relation to Arclength1:11
- Lecture Example 11:46
- Lecture Example 24:29
- Lecture Example 39:34
- Additional Example 4-1
- Additional Example 5-2

Hydrostatic Pressure

24m 37s

- Intro0:00
- Important Equation0:09
- Main Idea0:12
- Different Forces0:45
- Weight Density Constant1:10
- Variables (Depth and Width)2:21
- Lecture Example 13:28
- Additional Example 2-1
- Additional Example 3-2

Center of Mass

25m 39s

- Intro0:00
- Important Equation0:07
- Main Idea0:25
- Centroid1:00
- Area1:28
- Lecture Example 11:44
- Lecture Example 26:13
- Lecture Example 310:04
- Additional Example 4-1
- Additional Example 5-2

III. Parametric Functions

Parametric Curves

22m 26s

- Intro0:00
- Important Equations0:05
- Slope of Tangent Line0:30
- Arc length1:03
- Lecture Example 11:40
- Lecture Example 24:23
- Lecture Example 38:38
- Additional Example 4-1
- Additional Example 5-2

Polar Coordinates

30m 59s

- Intro0:00
- Important Equations0:05
- Polar Coordinates in Calculus0:42
- Area0:58
- Arc length1:41
- Lecture Example 12:14
- Lecture Example 24:12
- Lecture Example 310:06
- Additional Example 4-1
- Additional Example 5-2

IV. Sequences and Series

Sequences

31m 13s

- Intro0:00
- Definition and Theorem0:05
- Monotonically Increasing0:25
- Monotonically Decreasing0:40
- Monotonic0:48
- Bounded1:00
- Theorem1:11
- Lecture Example 11:31
- Lecture Example 211:06
- Lecture Example 314:03
- Additional Example 4-1
- Additional Example 5-2

Series

31m 46s

- Intro0:00
- Important Definitions0:05
- Sigma Notation0:13
- Sequence of Partial Sums0:30
- Converging to a Limit1:49
- Diverging to Infinite2:20
- Geometric Series2:40
- Common Ratio2:47
- Sum of a Geometric Series3:09
- Test for Divergence5:11
- Not for Convergence6:06
- Lecture Example 18:32
- Lecture Example 210:25
- Lecture Example 316:26
- Additional Example 4-1
- Additional Example 5-2

Integral Test

23m 26s

- Intro0:00
- Important Theorem and Definition0:05
- Three Conditions0:25
- Converging and Diverging0:51
- P-Series1:11
- Lecture Example 12:19
- Lecture Example 25:08
- Lecture Example 36:38
- Additional Example 4-1
- Additional Example 5-2

Comparison Test

22m 44s

- Intro0:00
- Important Tests0:01
- Comparison Test0:22
- Limit Comparison Test1:05
- Lecture Example 11:44
- Lecture Example 23:52
- Lecture Example 36:01
- Lecture Example 410:04
- Additional Example 5-1
- Additional Example 6-2

Alternating Series

25m 26s

- Intro0:00
- Main Theorems0:05
- Alternation Series Test (Leibniz)0:11
- How It Works0:26
- Two Conditions0:46
- Never Use for Divergence1:12
- Estimates of Sums1:50
- Lecture Example 13:19
- Lecture Example 24:46
- Lecture Example 36:28
- Additional Example 4-1
- Additional Example 5-2

Ratio Test and Root Test

33m 27s

- Intro0:00
- Theorems and Definitions0:06
- Two Common Questions0:17
- Absolutely Convergent0:45
- Conditionally Convergent1:18
- Divergent1:51
- Missing Case2:02
- Ratio Test3:07
- Root Test4:45
- Lecture Example 15:46
- Lecture Example 29:23
- Lecture Example 313:13
- Additional Example 4-1
- Additional Example 5-2

Power Series

38m 36s

- Intro0:00
- Main Definitions and Pattern0:07
- What Is The Point0:22
- Radius of Convergence Pattern0:45
- Interval of Convergence2:42
- Lecture Example 13:24
- Lecture Example 210:55
- Lecture Example 314:44
- Additional Example 4-1
- Additional Example 5-2

V. Taylor and Maclaurin Series

Taylor Series and Maclaurin Series

30m 18s

- Intro0:00
- Taylor and Maclaurin Series0:08
- Taylor Series0:12
- Maclaurin Series0:59
- Taylor Polynomial1:20
- Lecture Example 12:35
- Lecture Example 26:51
- Lecture Example 311:38
- Lecture Example 417:29
- Additional Example 5-1
- Additional Example 6-2

Taylor Polynomial Applications

50m 50s

- Intro0:00
- Main Formulas0:06
- Alternating Series Error Bound0:28
- Taylor's Remainder Theorem1:18
- Lecture Example 13:09
- Lecture Example 29:08
- Lecture Example 317:35
- Additional Example 4-1
- Additional Example 5-2

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For more information, please see full course syllabus of College Calculus: Level II

For more information, please see full course syllabus of College Calculus: Level II

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1 answer

Last reply by: Dr. William Murray

Thu Aug 4, 2016 6:01 PM

Post by Peter Ke on July 27, 2016

For example 3, you let u = 3tan(theta). Where did that idea come from and why?

1 answer

Last reply by: Dr. William Murray

Sat Feb 27, 2016 10:36 AM

Post by Mohsin Alibrahim on February 25, 2016

Dr Murray,

In example 3, how come du = [3sec^2(x)] while u = 9sec^2(x) ?

Thanks for the wonderful lecture

3 answers

Last reply by: Dr. William Murray

Fri Apr 10, 2015 1:07 PM

Post by Luvivia Chang on April 7, 2015

Hello Dr. Murray, I have a question about exemple 1

i think the square root of (1-sin^2(theta))=the square root of cos^2(theta)= the "absolute value of cos(theta)" instead of just cos (theta)

Am I correct? Thanks

2 answers

Last reply by: Dr. William Murray

Thu Feb 19, 2015 3:48 PM

Post by Thadeus McNamara on February 18, 2015

Dr. Murray, I am taking the Calc BC Exam. Are there any lectures in this College Calculus II playlist that I can skip because they are not a part of the exam? Thanks

2 answers

Last reply by: Dr. William Murray

Thu Feb 19, 2015 3:48 PM

Post by Thadeus McNamara on February 18, 2015

@4:51, isn't the integral of 2/3sintheta equal to -2/3costheta? You put the integral as 2/3costheta.

3 answers

Last reply by: Dr. William Murray

Wed Aug 27, 2014 9:46 AM

Post by brody wagerson on August 24, 2014

I believe there is an error in lecture example 4, on the last step to simplify the answer you cannot divide x^2-6x+9 by 9 to get x^2-6x, because they are addition and not multiplication. You must multiply out what is under the square root sign to get 9-(x^2-6x+9) the 9's will then cancel and give you 6x-x^2 under the square root just like in the beginning of the question, but without taking out an extra 1/3 to divide. Sorry if my explanation does not make sense I wrote this quickly. but i believe the correct answer to example 4 if I am correct will be 9/2[arcsin(x-3/3)+(x-3/3)*sqrt(6x-x^2)]+C. No stranger to criticism though so let me know if I made a mistake and professor Murray is right.

1 answer

Last reply by: Dr. William Murray

Tue Aug 5, 2014 3:18 PM

Post by Pedro Valdericeda on July 21, 2014

Thank you Prof. Murray. You are really good.

1 answer

Last reply by: Dr. William Murray

Tue Jun 17, 2014 12:03 PM

Post by ahmed ahmed on June 9, 2014

thank you professor! you are awesome!

1 answer

Last reply by: Dr. William Murray

Mon Mar 17, 2014 1:13 PM

Post by Peidong, He on March 16, 2014

WANNA SAY THANK YOU!

1 answer

Last reply by: Dr. William Murray

Fri Aug 23, 2013 11:52 AM

Post by Charles Zhou on August 21, 2013

This is absolutely a nice video! It does help me a lot. I just have a question. I'm a high school student and I wonder is there any difference or relationship between AP calculus BC and this College Calculus Level II

1 answer

Last reply by: Dr. William Murray

Thu Dec 6, 2012 5:40 PM

Post by Samuel Bass on December 6, 2012

In problem #1 in the solution.

You converted cos(theta) = squar(1-sin^2(theta))

yes this would be correct if the cos(theta) was squared e.g. cos^2(theta)

but in this case the cos(theta) is not raised to the power 2.... then you can't use the pythagoream Identitie as you did in this case..... Im I correct? Cause I can't see this specific step you did possible. Please Explain.

1 answer

Last reply by: Dr. William Murray

Mon Oct 22, 2012 1:48 PM

Post by Keisha Baxter on October 21, 2012

He factored incorrectly in Lecture example 1... in about the third line when he factors out the 4 from sqrt(4 - 4 sin^2 x) he puts 2 sqrt (1 - sin^2 x). So all of the coefficients following that is incorrect

1 answer

Last reply by: Dr. William Murray

Thu Apr 18, 2013 12:24 PM

Post by SOUFIANE LAMOUNI on May 16, 2012

you are a great Professor , thank you

1 answer

Last reply by: Dr. William Murray

Thu Apr 18, 2013 12:23 PM

Post by Kumar Sandrasegaran on February 26, 2012

Very very interesting. Makes difficult calculations look simple

1 answer

Last reply by: Dr. William Murray

Thu Apr 18, 2013 12:20 PM

Post by Real Schiran on November 22, 2011

Very Nice, Thanks...........

1 answer

Last reply by: Dr. William Murray

Thu Apr 18, 2013 12:19 PM

Post by Jaspreet Singh on February 21, 2010

this is so much more helpful than school! thanks for being so clear

1 answer

Last reply by: Dr. William Murray

Thu Apr 18, 2013 12:19 PM

Post by lynette stevenson on February 17, 2010

excellent lecture