Integration 215
Step 3. Rewrite:
∫
u^2 /^3
du
2
=
1
2
∫
u^2 /^3 du.
Step 4. Integrate:
1
2
(
u^5 /^3
5 / 3
)
+C=
3 u^5 /^3
10
+C.
Step 5. Replaceu:
3 ( 2 x− 5 )^5 /^3
10
+C.
Step 6. Differentiate and Check:
(
3
10
)(
5
3
)
( 2 x− 5 )^2 /^3 (2)=( 2 x− 5 )^2 /^3.
Example 4
Evaluate
∫
x^2
(x^3 − 8 )^5
dx.
Step 1. Letu=x^3 −8.
Step 2. Differentiate:du= 3 x^2 dx⇒
du
3
=x^2 dx.
Step 3. Rewrite:
∫
1
u^5
du
3
=
1
3
∫
1
u^5
du=
1
3
∫
u−^5 du.
Step 4. Integrate:
1
3
(
u−^4
− 4
)
+C.
Step 5. Replaceu:
1
− 12
(
x^3 − 8
)− 4
+Cor
− 1
12 (x^3 − 8 )^4
+C.
Step 6. Differentiate and Check:
(
−
1
12
)
(− 4 )
(
x^3 − 8
)− 5 (
3 x^2
)
=
x^2
(x^3 − 8 )^5
.
U-Substitution and Trigonometric Functions
Example 1
Evaluate
∫
sin 4xdx.
Step 1. Letu= 4 x.
Step 2. Differentiate:du= 4 dxor
du
4
=dx.
Step 3. Rewrite:
∫
sinu
du
4
=
1
4
∫
sinudu.
Step 4. Integrate:
1
4
(−cosu)+C=−
1
4
cosu+C.
Step 5. Replaceu:−
1
4
cos( 4 x)+C.
Step 6. Differentiate and Check:
(
−
1
4
)
(−sin 4x)( 4 )=sin 4x.