220 STEP 4. Review the Knowledge You Need to Score High
Example 6
Evaluate
∫
ex
ex+ 1
dx.
Step 1. Letu=ex+1.
Step 2. Differentiate:du=exdx.
Step 3. Rewrite:
∫
1
u
du.
Step 4. Integrate: ln|u|+C.
Step 5. Replaceu:ln
∣∣
ex+ 1
∣∣
+C.
Step 6. Differentiate and Check:
1
ex+ 1
·ex=
ex
ex+ 1
.
Example 7
Evaluate
∫
xe^3 x^2 dx.
Step 1. Letu= 3 x^2.
Step 2. Differentiate:du= 6 xdx⇒
du
6
=xdx.
Step 3. Rewrite:
∫
eu
du
6
=
1
6
∫
eudu.
Step 4. Integrate:
1
6
eu+C.
Step 5. Replaceu:
1
6
e^3 x^2 +C.
Step 6. Differentiate and Check:
1
6
(
e^3 x^2
)
( 6 x)=xe^3 x^2.
Example 8
Evaluate
∫
5 (^2 x)dx.
Step 1. Letu= 2 x.
Step 2. Differentiate:du= 2 dx⇒
du
2
=dx.
Step 3. Rewrite:
∫
5 u
du
2
=
1
2
∫
5 udu.
Step 4. Integrate:
1
2
( 5 u)/ln 5+C= 5 u/(2 ln 5)+C.
Step 5. Replaceu:
52 x
2ln5
+C.
Step 6. Differentiate and Check: (5^2 x)(2) ln 5/(2 ln 5)= 52 x.