# College Calculus: Level II Integration by Parts

Section 1: Advanced Integration Techniques: Lecture 1 | 24:52 min

In this lecture we are going to talk about Integration by Parts. First, we will introduce the main equation for integration by parts. We are going to see where it come from. The point of integration by parts is that we will be given a hard integral to solve. What we are going to do, is take the given integral and split it up into two parts, a U part and a dV part. Then we will invoke this formula to convert it into UV minus the integral of VdU, and if we do that right, then the second integral that we get will be an easier integral. We can finish the problem by doing that easier integral.

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

Fri Oct 7, 2016 9:56 AM

Post by selam berhe on October 5, 2016

Antibiotic pharmacokinetics An antibiotic tablet is taken and t hours later the concentration in the bloodstream

is C(t)= 3(e^-0.8t -e^-1.2t)

where C is measured in mgymL.

(a) What is the maximum concentration of the antibiotic

and when does it occur?

(b) Calculate integral of 0 to 2 C(t) dt and interpret your result.

(c) Calculate integral of 0 to infinity of C(t) and explain its meaning.

`

0 Cstd dt and explain its meaning.

3 answers

Last reply by: Dr. William Murray

Mon Jun 8, 2015 5:08 PM

Post by Joshua Bowen on May 11, 2015

How do we recognize a Intergration by part quickly?

1 answer

Last reply by: Dr. William Murray

Mon Aug 18, 2014 11:04 AM

Post by ahmed ahmed on August 16, 2014

Thank you from the bottom of my heart ... you are of a great help and i got A on my calculus-2 summer class... i am so happy... thank you again and again and forever...

1 answer

Last reply by: Dr. William Murray

Tue Feb 11, 2014 4:16 PM

Post by Silvia Gonzalez on February 3, 2014

Hi, thank you for taking the time to prepare such good lectures.

I am doing the practice exercise integral from 0 to n of (x-1)sinx.dx and I do not understand how you go from step 8 to 9, can you help me?

1 answer

Last reply by: Dr. William Murray

Sun Nov 3, 2013 6:23 PM

Post by Christian Fischer on November 3, 2013

Hi Professor, Thank you for a great video. There is one thing I don't understand: If u=x then du=dx but if u=x^2 then du=2x

To me it looks like we differentiated x^2 to get du=2x but did nothing to x to get du=dx? if we had differentiated x we would get du=d*1 What is the explanation for that?

Have a great day

Christian

2 answers

Last reply by: Dr. William Murray

Fri Aug 23, 2013 11:48 AM

Post by Maximillian Lanander on August 17, 2013

I think you made a boo boo. Not sure. If u is x^2 then du surely = x^2dx ?

3 answers

Last reply by: Dr. William Murray

Fri Mar 11, 2016 9:41 AM

Post by Ramez Hajelsawi on July 7, 2013

That shortcut blew my mind, I went back and redid all the problems that it works for again just to prove it to myself. Fantastic lecture!

1 answer

Last reply by: Dr. William Murray

Tue Apr 23, 2013 7:49 PM

Post by Hye Goo Yoon on April 23, 2013

integration of [(3X^2)/7^3]e^-(x/7)^3 dx 0<X<âˆž

At first I have used by parts method but it was a never ending integration.

I have found a formula from the table of integrals, it still gave me unfinished integral. How can I effectively solve this kind of integration by parts? Is there any ways for me to easily approach this kind of integration? Since the range is given,

Somehow I have ended up learning about Gamma function. I am very confused now.

Thank you

1 answer

Last reply by: Dr. William Murray

Fri Jan 18, 2013 5:30 PM

Post by Abdelrahman Megahed on January 17, 2013

How come LIATE doesn't work for example 3?

1 answer

Last reply by: Dr. William Murray

Fri Jan 18, 2013 5:30 PM

Post by Shehan Gunasekara on May 16, 2012

WOW!! This is so goood!! you are better than my lecturer

1 answer

Last reply by: Dr. William Murray

Fri Jan 18, 2013 5:33 PM

Post by Emudiare Sowho on May 7, 2012

How do we get the practice problems?

1 answer

Last reply by: Dr. William Murray

Fri Jan 18, 2013 5:34 PM

Post by Kathy Hodge on November 7, 2011

Thanks. Helped me understand and apply this independently, without having to look at aids, for the first time, despite some previous attempts.

3 answers

Last reply by: Dr. William Murray

Fri Jan 18, 2013 5:36 PM

Post by Maimouna Louche on June 7, 2011

Why does it stop and start over? It stops at the "Shortcut: Tabular Integration".

3 answers

Last reply by: Dr. William Murray

Fri Jan 18, 2013 5:37 PM

Post by Maimouna Louche on June 7, 2011

what is the difference between derivatives and integrals?

1 answer

Last reply by: Dr. William Murray

Fri Jan 18, 2013 5:38 PM

Post by Anthony Simon on April 20, 2010

This is a great video, i can't wait for those practice problems.

1 answer

Last reply by: Dr. William Murray

Fri Jan 18, 2013 5:41 PM

Post by Stephan Maric on March 13, 2010

Excellent video instruction.

My only question is whether or not you have access to supplemental problems. This would really help me practice out the problems.