5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book May 8, 2018 13:52

Integration 241

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

4ln5

=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.

11.3 Rapid Review

1. Evaluate

∫

1

x^2

dx.

Answer:Rewrite as

∫

x−^2 dx=

x−^1

− 1

+C=−

1

x

+C.

2. Evaluate

∫

x^3 − 1

x

dx.

Answer:Rewrite as

∫ (

x^2 −

1

x

)

dx=

x^3

3

−ln|x|+C.

3. Evaluate

∫

x

√

x^2 − 1 dx.