3. Integration by Parts and the Product Rule
› “Every differentiation rule has a corresponding integration
rule”
– Chain Rule (differentiation) Substitution Rule (Integration)
› Integration by Parts (integration) is correspondent to the
Product Rule (differentiation)
4. Integration by Parts and the Product Rule
[ ( ) ( )] ( ) ( ) ( ) ( )
( ) ( ) [ ( ) ( ) ( ) ( )]
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
d
f x g x f x g x g x f x
dx
f x g x f x g x g x f x dx
f x g x f x g x dx g x f x dx
f x g x dx f x g x g x f x dx
The product rule
(Equation 1), through
manipulation, results in
Integration by Parts
(Equation 4).
6. The Formula for Integration by Parts
Through substitutions of and , the above
equation ‘simplifies’ to the formula for Integration by Parts:
( ) ( ) ( ) ( ) ( ) ( )f x g x dx f x g x g x f x dx
( )u f x ( )v g x
udv uv vdu
8. Using Integration by Parts
› The aim of using Integration by Parts
– Obtain a simpler integral that is easier to evaluate than the
original integral
› Integration by Parts is not always useful
– Attempting Integration by Parts may reveal that the resulting
integral on the Right Hand Side is more complicated and difficult
to evaluate than the original integral
› Integration by Parts can be nested
– It’s important to note that Integration by Parts can be used on
the resulting integral from using Integration by Parts
9. Using Integration by Parts
A
• Identify Functions
B
• Delegate
C
• Evaluate
› Identify the two parts of
the original integral
› To delegate each part to
and , a good rule of
thumb is that the function
that becomes simpler when
differentiated should be
› Using the formula, evaluate
the integral
u
dv
u
10. Using Integration by Parts
› It can be helpful to use the pattern as shown.
...u
...du
...dv
...v
12. Evaluate ln xdx
lnu x
1
du dx
x
dv dx
v x
Recalling the formula for Integration by Parts: . Note that the above
integral is quite crucial and the answer should be memorized if possible.
udv uv vdu
13. Evaluate ln xdx
ln ln
ln
ln
dx
xdx x x x
x
x x dx
x x x C
This is an extremely simple and
straightforward use of
Integration by Parts, but is
sufficient to demonstrate its
power. The given integral would
not very difficult and
complicated to solve without
utilizing Integration by Parts
15. Evaluate sinx
e xdx
sinu x
cosdu xdx
x
dv e dx
x
v e
Recalling the formula for Integration by Parts: . Note that in the above
integral, neither parts become simpler when differentiated. However, in hopes of avoiding
negatives, .
udv uv vdu
sinu x
16. Evaluate
sin sin cosx x x
e xdx e x e xdx
Using the Formula for Integration of Parts is meant to result in an
integral that is simpler to evaluate than the original. In this case, we
result in an integral with equal complexity. However, since the
resulting integral is not more complex, we apply the Formula for
Integration of Parts on the resulting integral.
sinx
e xdx
17. Evaluate
cosu x
sindu xdx
x
dv e dx
x
v e
Recalling the formula for Integration by Parts: .udv uv vdu
cosx
e xdx
18. Evaluate
cos cos sinx x x
e xdx e x e xdx
The resulting integral this time is, again, of equal complexity. However,
notice that it is equal to the original integral. Using this, we can
evaluate the original integral and break from this seemingly never-
ending loop.
cosx
e xdx
19. Evaluate
sin sin cos
sin ( cos sin )
sin cos sin
1
( sin cos )
2
x x x
x x x
x x x
x x
e xdx e x e x
e x e x e xdx
e x e x e xdx
e x e x C
In this example, utilizing Integration by Parts initially seemed to be
irrelevant, as the resulting integral was equal in complexity to the
original integral. However, “nesting” Integration by Parts allowed to
algebraically manipulate and ultimately solve for the original integral.
sinx
e xdx
