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Midterm

MATH 2B Study Guide - Midterm Guide: Product RulePremium


Department
Mathematics
Course Code
MATH 2B
Professor
All
Study Guide
Midterm

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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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