# College Calculus: Level II Trigonometric Substitutions

Section 1: Advanced Integration Techniques: Lecture 3 | 30:09 min

In this video we are going to talk about Trigonometric Substitutions. There are three main equations that we have for trigonometric substitution. First we are going to list them. The idea is that if we see any one of these three forms in an integral, then we can do what is called a Trigonometric Substitution to convert out integral into a trigonometric integral. Then we can use some of the trigonometric techniques that we learned in the previous lecture to solve the integral. The best way to learn this is to do lots of examples, so we are going to do some of them.

Professor Murray

Trigonometric Substitutions

Slide Duration:Table of Contents

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1 answer

Last reply by: Dr. William Murray

Mon Feb 4, 2019 2:25 PM

Post by Hessa on February 1, 2019

how to solve this question using trig sub

the integral of x(1-x^4) ^1/2?

1 answer

Last reply by: Dr. William Murray

Mon Jan 28, 2019 11:43 AM

Post by Hessa on January 27, 2019

I did not understand example 3

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