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
- Evaluate
∫
1
x^2
dx.
Answer:Rewrite as
∫
x−^2 dx=
x−^1
− 1
+C=−
1
x
+C.
- Evaluate
∫
x^3 − 1
x
dx.
Answer:Rewrite as
∫ (
x^2 −
1
x
)
dx=
x^3
3
−ln|x|+C.
- Evaluate
∫
x
√
x^2 − 1 dx.