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