Section 8.2 – Integration by Parts
Dec 23, 2015
Find the ErrorThe following is an example of a student response. How can you tell the final answer is incorrect? Where did the student make an error?
Evaluate:
This is not the antiderivative of since
2
cos cos
1sin
2
x x dx x dx x dx
x x C
The integral of a product is not equal to product of the integrals.
This should remind us of the Product Rule. Is there a way to use the Product Rule to investigate the
antiderivative of a product?
Integration by Parts: An Explanation
When u and v are differentiable functions of x:
d dv dudx dx dxuv u v The Product
Rule tells us…
d dv dudx dx dxuv dx u dx v dx If we integrate
both sides…
uv u dv v du If we simplify the integrals…
u dv uv v du If we solve for one of the integrals…
Integration by Parts
When evaluating the integral and f (x)dx=u dv (with u and v being differentiable functions of x), then, the following holds:
u dv uv v du Rewrite the function into
the product of u and dv.
The integral equals…
u times the antiderivative
of dv.
The integral of the product of
the antiderivative of dv and the
derivative of u.
Integration by Parts: The Process
Since f (x)dx=u dv, success in using this important technique depends on being able to separate a given integral into parts u and dv so that…
a) dv can be integrated.
b) is no more difficult to calculate than the original integral.
The following does NOT always hold, but is very helpful:
Frequently, the derivative of u, or any higher order derivative, will be zero.
Example 1
Evaluate:
u Pick the u and dv.
dv x cos x dx
du Find du and v. v dx sin x
cosx x dx Apply the formula. sinx x sin x dx
sin cosx x x C
Differentiate. Integrate.
Example 2
Evaluate: u Pick the u
and dv.dv 2x xe dx
du Find du and v. v 2xdx xe
2 xx e dx Apply the formula.
2 xx e 2 xxe dx Differentiate. Integrate.
You may need to apply
Integration by Parts Again.
u Pick the u and dv.
dv 2x xe dx
du Find du and v. v 2dx xe
2 xx e Apply the formula. 2 xxe 2 xe dx
2 2 2x x xx e xe e C
2 xxe dx
Example 3
Evaluate: Since, multiple Integration by Parts are needed, a Tabular Method is a
convenient method for organizing repeated Integration by parts.
Repeated Differentiation Repeated Integration3x x23 1x 6x
60
1 21x 3 22
3 1x 5 24
15 1x 7 28
105 1x 9 216
945 1x
+–+
–
3 2323 1x x x 5 224
15 3 1 1x x 7 21635 1x x 9 232
315 1x C
Must get 0.
Start with +Alternate
Find the sum of the products of each diagonal:
Differentiate the u.
Integrate the dv.
Connect the
diagonals.
Notice the cubic function will go to zero. So it is a good choice for u.
Example 4
Evaluate:
u Pick the u and dv.
dv x ln x dx
du Find du and v. v dx ??
This was a bad choice for u and dv.
Differentiate. Integrate.
Example 4: Second Try
Evaluate:
u Pick the u and dv.
dv ln x x dx
du Find du and v. v 1
x dx 212 x
lnx x dx Apply the formula.
212 lnx x 1
2 x dx 2 21 1
2 4lnx x x C
Differentiate. Integrate.
Try the opposite this time.
Example 5
Evaluate:
1 ln x dxIf there is only one function, rewrite the integral so there is two.
u Pick the u and dv.
dv ln x 1dx
du Find du and v. v 1
x dx x
ln x dx Apply the formula. lnx x 1dx
lnx x x C
Differentiate. Integrate.
Example 6
Evaluate: u Pick the u and dv. dv cos x xe dx
du Find du and v. v sin x dx xecosxe x dx Apply the
formula. cosxe x sinxe x dx You may need
to apply Integration by Parts Again.
cos cosx xe x dx e x Apply the formula. sinxe x cosxe x dx
1 12 2cos cos sinx x xe x dx e x e x C
cos cos sinx x xe x dx e x e x dx sinxe x dxu Pick the u and dv. dv sin x xe dx
du Find du and v. v cos x dx xe
2 cos cos sinx x xe x dx e x e x If you see the integral you are
trying to find, solve for it.
Example 7
Evaluate: 1 1
01 tan x dxIf there is only one function, rewrite the
integral so there is two.
u Pick the u and dv.
dv 1tan x 1dx
du Find du and v. v 2
11 x
dx
x1 1
01 tan x dx Apply the
formula. 11
0tanx x
2
1
10
xx
dx
1
214 2 0
ln 1 x 1
4 2 ln 2
Integration by Parts: Helpful Acronym
When deciding which product to make u, choose the function whose category occurs earlier in the list below. Then take dv to be the rest of the integrand.
L
I
A
T
E
ogarithmicnverse trigonometric
lgebraic
rigonometric
xponential