218 STEP 4. Review the Knowledge You Need to Score High
U-Substitution and Logarithmic and Exponential Functions
Example 1
Evaluate
∫
x^3
x^4 − 1
dx.
Step 1. Letu=x^4 −1.
Step 2. Differentiate:du= 4 x^3 dx⇒
du
4
=x^3 dx.
Step 3. Rewrite:
∫
1
u
du
4
=
1
4
∫
1
u
du.
Step 4. Integrate:
1
4
ln|u|+C.
Step 5. Replaceu:
1
4
ln
∣
∣x^4 − 1
∣
∣+C.
Step 6. Differentiate and Check:
(
1
4
)
1
x^4 − 1
(
4 x^3
)
=
x^3
x^4 − 1
.
Example 2
Evaluate
∫
sinx
cosx+ 1
dx.
Step 1. Letu=cosx+1.
Step 2. Differentiate:du=−sinxdx⇒−du=sinxdx.
Step 3. Rewrite:
∫
−du
u
=−
∫
du
u
.
Step 4. Integrate:−ln|u|+C.
Step 5. Replaceu:−ln
∣∣
cosx+ 1
∣∣
+C.
Step 6. Differentiate and Check:−
(
1
cosx+ 1
)
(−sinx)=
sinx
cosx+ 1
.
Example 3
Evaluate
∫
x^2 + 3
x− 1
dx.
Step 1. Rewrite:
x^2 + 3
x− 1
=x+ 1 +
4
x− 1
by dividing (x^2 + 3) by (x−1).
∫
x^2 + 3
x− 1
dx=
∫ (
x+ 1 +
4
x− 1
)
dx=
∫
(x+ 1 )dx+
∫
4
x− 1
dx
=
x^2
2
+x+ 4
∫
1
x− 1
dx
Letu=x−1.
Step 2. Differentiate:du=dx.