216 STEP 4. Review the Knowledge You Need to Score High
Example 2
Evaluate
∫
3
(
sec^2 x
)√
tanxdx.
Step 1. Letu=tanx.
Step 2. Differentiate:du=sec^2 xdx.
Step 3. Rewrite: 3
∫
(tanx)^1 /^2 sec^2 xdx= 3
∫
u^1 /^2 du.
Step 4. Integrate: 3
u^3 /^2
3 / 2
+C= 2 u^3 /^2 +C.
Step 5. Replaceu: 2(tanx)^3 /^2 +Cor 2 tan^3 /^2 x+C.
Step 6. Differentiate and Check:( 2 )
(
3
2
)(
tan^1 /^2 x
)(
sec^2 x
)
= 3
(
sec^2 x
)√
tanx.
Example 3
Evaluate
∫
2 x^2 cos
(
x^3
)
dx.
Step 1. Letu=x^3.
Step 2. Differentiate:du= 3 x^2 dx⇒
du
3
=x^2 dx.
Step 3. Rewrite: 2
∫ [
cos
(
x^3
)]
x^2 dx= 2
∫
cosu
du
3
=
2
3
∫
cosudu.
Step 4. Integrate:
2
3
sinu+C.
Step 5. Replaceu:
2
3
sin
(
x^3
)
+C.
Step 6. Differentiate and Check:
2
3
[
cos
(
x^3
)]
3 x^2 = 2 x^2 cos
(
x^3
)
.
TIP • Remember that the area of a semicircle is^1
2
πr^2. Do not forget the
1
2
. If the cross
sections of a solid are semicircles, the integral for the volume of the solid will involve(
1
2
) 2
which is
1
4
.
U-Substitution and Inverse Trigonometric Functions
Example 1
Evaluate
∫
√dx
9 − 4 x^2
.
Step 1. Letu= 2 x.
Step 2. Differentiate:du= 2 dx;
du
2
=dx.