8.2 Integration by Parts
Objectives:
1.
Integration by Parts
2.
Column integration
Quick Review: the Product Rule
If
?(?)
???
?(?)
are both differentiable and
?(?) = ?(?)?(?)
, then
?
??
?(?) =
?
??
[?(?)?(?)] = ?(?)
?
??
[?(?)] + ?(?)
?
??
[?(?)]
•
Write the rule in prime notation:
Example 1
:
? = (?
2
+ ? + 3)(?
x
− 5)
Find
??
??
Simplify the result.
Algebra Review
:
?
?
= ? ∙ ?
−1
, ??𝑟 ? ≠ 0
Example 2
: Rewrite each as a product:
? =
ln ?
?
? =
?
2
?
3𝑥
? =
5
?
6
? =
1
√?
3

(2) Integration by Parts
Integration by Parts
∫ ?(𝒙)?′(𝒙) 𝒅𝒙 = ?(𝒙)?(𝒙) − ∫ ?(𝒙)?′(𝒙)𝒅𝒙
Or letting
? = ?(?)
and
? = ?(?),
we
get
∫ ?𝒅? = ?? − ∫ ?𝒅?
Tips:
1. Choose
? = ?(?)
to be a function that becomes simpler when differentiated and
?? =
?′(?)??
to be the part that can be readily integrated to give back v.
2. Check v by differentiating it before using the formula.
Experiences suggest the following rule: whichever function comes first in the list LATE is
to be u and the rest is dv.
L
: Logarithmic functions:
ln ?,
log
2
?
, etc
A
: Algebraic functions: polynomials
?
2
,
3?
7
,
?
2
+ 6
, etc
T
: Trigonometric functions:
sin ? ,
cos ?
, etc
E
: Exponential function:
?
?
,
13
?
, etc
Example 3
:
For each function, what will you choose for u?
What will be dv?