5 Steps to a 5 AP Calculus BC 2019

Integration 221

Example 9

Evaluate

∫

x^35 (x

(^4) )
dx.
Step 1. Letu=x^4.
Step 2. Differentiate:du= 4 x^3 dx⇒
du
4
=x^3 dx.
Step 3. Rewrite:
∫
5 u
du
4

=

1

4

∫

5 udu.

Step 4. Integrate:

1

4

( 5 u)/ln 5+C.

Step 5. Replaceu:

5 x^4

4ln5

+C.

Step 6. Differentiate and Check: 5(x

(^4) )(
4 x^3
)
ln 5/(4 ln 5)=x^35 (x
(^4) )
.
Example 10
Evaluate
∫
(sinπx)ecosπxdx.
Step 1. Letu=cosπx.
Step 2. Differentiate:du=−πsinπxdx;−
du
π
=sinπxdx.
Step 3. Rewrite:
∫
eu
(
−du
π
)
=−

1

π

∫

eudu.

Step 4. Integrate:−

1

π

eu+C.

Step 5. Replaceu:−

1

π

ecosπx+C.

Step 6. Differentiate and Check:−

1

π

(ecosπx)(−sinπx)π=(sinπx)ecosπx.

10.3 Techniques of Integration

Main Concepts:Integration by Parts, Integration by Partial Fractions

Integration by Parts

According to the product rule for differentiation

d

dx

(uv)=u

dv

dx

+v

du

dx

. Integrating

tells us thatuv=

∫

u

dv

dx

+

∫

v

du

dx

, and therefore

∫

u

dv

dx

=uv−

∫

v

du

dx

. To inte-

grate a product, careful identification of one factor asu and the other as

dv

dx

allows

the application of this rule for integration by parts. Choice of one factor to be u

(and therefore the other to bedv) is simpler if you remember the mnemonicLIPET.