Department

MathematicsCourse Code

MATH 2BProfessor

AllStudy Guide

MidtermThis

**preview**shows pages 1-2. to view the full**6 pages of the document.**Section 7.1 - Integration by Parts

What is the derivative of ()()?

●Product rule

●(()()) = ()() + ()()

●This kind of answer is difficult to see if you want to take the integral of something

●How can we rewrite it?

●

(()()) =

()() +

()()

○First take the integral

○()() =

()() +

()()

■ The integral of the left side is the same as ()()

○

()() = ()()

()()

■ Rearrange

●What does this mean?

○

=

○Replacing () , () , you can get this formula

○This is integration by parts

Example #1

●

○ = , which means that = 1

■ From the integral, replace with which parts look easiest to derive

○ = , then =

■ From the integral, replace with which parts look easiest to integrate

○

=

○

= ()

○=

○(1) +

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How do we know which to set to and which to set to ?

●LIATE

○Logs

○Inverse Logs

○Algebra

○Trig

○Exponentials

●Going down the list means that it is easier to take the integral of

○In other words, logs are the easiest to take the derivative of while exponentials

are the easiest to take the integral of

Example #2

●

(2)

○ = , so = 1

○ = (2), so =

(2)

■ Here, we have to use substitution to further integrate

■ = 2, (

) =

■

()

■

() =

(2)

■ =

(2)

○= (

(2))

(2)(1)

■ Plug in ,,

○

(((2))) +

(()

) +

○

(((2))) +

((2)) +

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