Section 8.2 – Integration by Parts
Feb 22, 2016
Section 8.2 – Integration by Parts
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
1 sin2
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 PartsWhen 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 ProcessSince 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 1Evaluate:
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 2Evaluate:
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
White Board ChallengeEvaluate:
3 2323 1x x x 5 224
15 3 1 1x x 7 21635 1x x 9 232
315 1x C
Example 3Evaluate:
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 6x60
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 4Evaluate:
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 TryEvaluate:
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 5Evaluate:
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 6Evaluate:
u Pick the u and dv. dv cos x xe dxdu 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 7Evaluate:
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
10xx
dx
121
4 2 0ln 1 x
14 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.LIATE
ogarithmicnverse trigonometric
lgebraicrigonometricxponential
White Board ChallengeEvaluate:
3 2432 1 1x x x C