Trigonometric Substitutions, College Calculus: Level II

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Trigonometric Substitutions

Main formula:

For , substitute .
For , substitute .
For , substitute .

Hints and tips:

Remember to make the substitution as well as the main substitution.
You can sometimes use trigonometric substitution even when you don’t have a square root. However, in these cases, you should only be using the tangent substitution. If you get one of the other ones, then you should have been able to factor the algebraic expression, and trigonometric substitution was not necessary.
If you have a linear term (x or u, as opposed to x² or u²), complete the square and make a substitution to eliminate the linear term before you make the trigonometric substitution.
If the coefficient of the quadratic term (x²) is negative, factor the negative out of all terms before completing the square.
After you make the trigonometric substitution, it becomes a trigonometric integral that you can handle using the techniques in the previous section.

Trigonometric Substitutions

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

Important Equations

Lecture Example 1

Lecture Example 2

Lecture Example 3

Additional Example 4

Additional Example 5

I. Advanced Integration Techniques
	Integration by Parts	24:52
	Integration of Trigonometric Functions	25:30
	Trigonometric Substitutions	30:09
	Partial Fractions	41:22
	Integration Tables	20:00
	Trapezoidal Rule, Midpoint Rule, Left/Right Endpoint Rule	22:36
	Simpson's Rule	21:08
	Improper Integration	44:18
II. Applications of Integrals, part 2
	Arclength	23:20
	Surface Area of Revolution	28:53
	Hydrostatic Pressure	24:37
	Center of Mass	25:39
III. Parametric Functions
	Parametric Curves	22:26
	Polar Coordinates	30:59
IV. Sequences and Series
	Sequences	31:13
	Series	31:46
	Integral Test	23:26
	Comparison Test	22:44
	Alternating Series	25:26
	Ratio Test and Root Test	33:27
	Power Series	38:36
V. Taylor and Maclaurin Series
	Taylor Series and Maclaurin Series	30:18
	Taylor Polynomial Applications	50:50

Mathematics: College Calculus: Level II

I. Advanced Integration Techniques
	Integration by Parts			24:52
		Intro		0:00
		Important Equation		0:07
			Where It Comes From (Product Rule)	0:20
			Why Use It?	0:35
		Lecture Example 1		1:24
		Lecture Example 2		3:30
		Shortcut: Tabular Integration		7:34
			Example	7:52
		Lecture Example 3		10:00
		Mnemonic: LIATE		14:44
			Ln, Inverse, Algebra, Trigonometry, e	15:38
		Additional Example 4		3:13
		Additional Example 5		4:13
	Integration of Trigonometric Functions			25:30
		Intro		0:00
		Important Equation		0:07
			Powers (Odd and Even)	0:19
			What To Do	1:03
		Lecture Example 1		1:37
		Lecture Example 2		3:12
		Half-Angle Formulas		6:16
			Both Powers Even	6:31
		Lecture Example 3		7:06
		Lecture Example 4		10:59
		Additional Example 5		2:37
		Additional Example 6		7:23
	Trigonometric Substitutions			30:09
		Intro		0:00
		Important Equations		0:06
			How They Work	0:35
			Example	1:45
			Remember: du and dx	2:50
		Lecture Example 1		3:43
		Lecture Example 2		10:01
		Lecture Example 3		12:04
		Additional Example 4		10:36
		Additional Example 5		3:21
	Partial Fractions			41:22
		Intro		0:00
		Overview		0:07
			Why Use It?	0:18
		Lecture Example 1		1:21
		Lecture Example 2		6:52
		Lecture Example 3		13:28
		Additional Example 4		8:47
		Additional Example 5		13:21
	Integration Tables			20:00
		Intro		0:00
		Using Tables		0:09
			Match Exactly	0:32
		Lecture Example 1		1:16
		Lecture Example 2		5:28
		Lecture Example 3		8:51
		Additional Example 4		3:05
		Additional Example 5		4:11
	Trapezoidal Rule, Midpoint Rule, Left/Right Endpoint Rule			22:36
		Intro		0:00
		Trapezoidal Rule		0:13
			Graphical Representation	0:20
			How They Work	1:08
			Formula	1:47
			Why a Trapezoid?	2:53
		Lecture Example 1		5:10
		Midpoint Rule		8:23
			Why Midpoints?	8:56
			Formula	9:37
		Lecture Example 2		11:22
		Left/Right Endpoint Rule		13:54
			Left Endpoint	14:08
			Right Endpoint	14:39
		Lecture Example 3		15:32
		Additional Example 4		2:56
		Additional Example 5		2:19
	Simpson's Rule			21:08
		Intro		0:00
		Important Equation		0:03
			Estimating Area	0:28
			Difference from Previous Methods	0:50
			General Principle	1:09
		Lecture Example 1		3:49
		Lecture Example 2		6:32
		Lecture Example 3		9:07
		Additional Example 4		2:56
		Additional Example 5		5:51
	Improper Integration			44:18
		Intro		0:00
		Horizontal and Vertical Asymptotes		0:04
			Example: Horizontal	0:16
			Formal Notation	0:37
			Example: Vertical	1:58
			Formal Notation	2:29
		Lecture Example 1		5:01
		Lecture Example 2		7:41
		Lecture Example 3		11:32
		Lecture Example 4		15:49
		Formulas to Remember		18:26
			Improper Integrals	18:36
		Lecture Example 5		21:34
		Lecture Example 6 (Hidden Discontinuities)		26:51
		Additional Example 7		2:57
		Additional Example 8		6:04
II. Applications of Integrals, part 2
	Arclength			23:20
		Intro		0:00
		Important Equation		0:04
			Why It Works	0:49
			Common Mistake	1:21
		Lecture Example 1		2:14
		Lecture Example 2		6:26
		Lecture Example 3		10:49
		Additional Example 4		5:16
		Additional Example 5		3:25
	Surface Area of Revolution			28:53
		Intro		0:00
		Important Equation		0:05
			Surface Area	0:38
			Relation to Arclength	1:11
		Lecture Example 1		1:46
		Lecture Example 2		4:29
		Lecture Example 3		9:34
		Additional Example 4		5:00
		Additional Example 5		4:54
	Hydrostatic Pressure			24:37
		Intro		0:00
		Important Equation		0:09
			Main Idea	0:12
			Different Forces	0:45
			Weight Density Constant	1:10
			Variables (Depth and Width)	2:21
		Lecture Example 1		3:28
		Additional Example 2		9:24
		Additional Example 3		4:32
	Center of Mass			25:39
		Intro		0:00
		Important Equation		0:07
			Main Idea	0:25
			Centroid	1:00
			Area	1:28
		Lecture Example 1		1:44
		Lecture Example 2		6:13
		Lecture Example 3		10:04
		Additional Example 4		4:05
		Additional Example 5		6:48
III. Parametric Functions
	Parametric Curves			22:26
		Intro		0:00
		Important Equations		0:05
			Slope of Tangent Line	0:30
			Arc length	1:03
		Lecture Example 1		1:40
		Lecture Example 2		4:23
		Lecture Example 3		8:38
		Additional Example 4		5:00
		Additional Example 5		4:22
	Polar Coordinates			30:59
		Intro		0:00
		Important Equations		0:05
			Polar Coordinates in Calculus	0:42
			Area	0:58
			Arc length	1:41
		Lecture Example 1		2:14
		Lecture Example 2		4:12
		Lecture Example 3		10:06
		Additional Example 4		8:38
		Additional Example 5		9:02
IV. Sequences and Series
	Sequences			31:13
		Intro		0:00
		Definition and Theorem		0:05
			Monotonically Increasing	0:25
			Monotonically Decreasing	0:40
			Monotonic	0:48
			Bounded	1:00
			Theorem	1:11
		Lecture Example 1		1:31
		Lecture Example 2		11:06
		Lecture Example 3		14:03
		Additional Example 4		6:11
		Additional Example 5		7:24
	Series			31:46
		Intro		0:00
		Important Definitions		0:05
			Sigma Notation	0:13
			Sequence of Partial Sums	0:30
			Converging to a Limit	1:49
			Diverging to Infinite	2:20
		Geometric Series		2:40
			Common Ratio	2:47
			Sum of a Geometric Series	3:09
		Test for Divergence		5:11
			Not for Convergence	6:06
		Lecture Example 1		8:32
		Lecture Example 2		10:25
		Lecture Example 3		16:26
		Additional Example 4		3:47
		Additional Example 5		4:26
	Integral Test			23:26
		Intro		0:00
		Important Theorem and Definition		0:05
			Three Conditions	0:25
			Converging and Diverging	0:51
			P-Series	1:11
		Lecture Example 1		2:19
		Lecture Example 2		5:08
		Lecture Example 3		6:38
		Additional Example 4		6:18
		Additional Example 5		4:49
	Comparison Test			22:44
		Intro		0:00
		Important Tests		0:01
			Comparison Test	0:22
			Limit Comparison Test	1:05
		Lecture Example 1		1:44
		Lecture Example 2		3:52
		Lecture Example 3		6:01
		Lecture Example 4		10:04
		Additional Example 5		4:11
		Additional Example 6		4:20
	Alternating Series			25:26
		Intro		0:00
		Main Theorems		0:05
			Alternation Series Test (Leibniz)	0:11
			How It Works	0:26
			Two Conditions	0:46
			Never Use for Divergence	1:12
			Estimates of Sums	1:50
		Lecture Example 1		3:19
		Lecture Example 2		4:46
		Lecture Example 3		6:28
		Additional Example 4		5:04
		Additional Example 5		9:44
	Ratio Test and Root Test			33:27
		Intro		0:00
		Theorems and Definitions		0:06
			Two Common Questions	0:17
			Absolutely Convergent	0:45
			Conditionally Convergent	1:18
			Divergent	1:51
			Missing Case	2:02
		Ratio Test		3:07
		Root Test		4:45
		Lecture Example 1		5:46
		Lecture Example 2		9:23
		Lecture Example 3		13:13
		Additional Example 4		9:13
		Additional Example 5		8:07
	Power Series			38:36
		Intro		0:00
		Main Definitions and Pattern		0:07
			What Is The Point	0:22
			Radius of Convergence Pattern	0:45
			Interval of Convergence	2:42
		Lecture Example 1		3:24
		Lecture Example 2		10:55
		Lecture Example 3		14:44
		Additional Example 4		8:39
		Additional Example 5		7:46
V. Taylor and Maclaurin Series
	Taylor Series and Maclaurin Series			30:18
		Intro		0:00
		Taylor and Maclaurin Series		0:08
			Taylor Series	0:12
			Maclaurin Series	0:59
		Taylor Polynomial		1:20
		Lecture Example 1		2:35
		Lecture Example 2		6:51
		Lecture Example 3		11:38
		Lecture Example 4		17:29
		Additional Example 5		2:50
		Additional Example 6		4:41
	Taylor Polynomial Applications			50:50
		Intro		0:00
		Main Formulas		0:06
			Alternating Series Error Bound	0:28
			Taylor's Remainder Theorem	1:18
		Lecture Example 1		3:09
		Lecture Example 2		9:08
		Lecture Example 3		17:35
		Additional Example 4		5:56
		Additional Example 5		13:50

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Trigonometric Substitutions

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Features Overview

Trigonometric Substitutions

Trigonometric Substitutions

Mathematics: College Calculus: Level II

Related Books

Related Books

	Calculus by Larson, 9th Edition Authors: Ron Larson, Bruce H. Edwards ISBN: 547167024 Publisher: Brooks Cole Year: 2009 The book is filled with examples and detailed explanations. Capstone exercises are also included for each section and synthesize the main concepts into a single example.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 1 Author: Ron Larson ISBN: 547213093 Publisher: Brooks Cole Year: 2008 This solutions manual includes worked out solutions to every odd-numbered exercise in Calculus of a Single Variable, 9th edition.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 2 Authors: Ron Larson, Bruce H. Edwards ISBN: 547213107 Publisher: Brooks Cole Year: 2009 This solutions manual includes worked out solutions to every odd-numbered exercise in Multivariable Calculus, 9th edition.