224 STEP 4. Review the Knowledge You Need to Score High
- Evaluate
∫
cos(2x)dx.
Answer:Letu= 2 xand obtain
1
2
sin 2x+C.
- Evaluate
∫
lnx
x
dx.
Answer:Letu=lnx;du=
1
x
dxand obtain
(lnx)^2
2
+C.
- Evaluate
∫
xex
2
dx.
Answer:Letu=x^2 ;
du
2
=xdxand obtain
ex^2
2
+C.
8.
∫
xcosxdx
Answer:Letu=x,du=dx,dv=cosxdx, andv=sinx,
then
∫
xcosxdx=xsinx−
∫
sinxdx=xsinx+cosx+C.
9.
∫
5
(x+3)(x−7)
dx
Answer:
∫
5
(x+3)(x−7)
dx=
∫ (
− 1 / 2
x+ 3
+
1 / 2
x− 7
)
dx
=−
1
2
ln
∣∣
x+ 3
∣∣
+
1
2
ln
∣∣
x− 7
∣∣
+C=
1
2
ln
∣∣
∣∣x−^7
x+ 3
∣∣
∣∣+C
10.5 Practice Problems
Evaluate the following integrals in problems
1 to 25. No calculators are allowed. (However,
you may use calculators to check your
results.)
1.
∫
(x^5 + 3 x^2 −x+1)dx
2.
∫ (√
x−
1
x^2
)
dx
3.
∫
x^3 (x^4 −10)^5 dx
4.
∫
x^3
√
x^2 + 1 dx
5.
∫
x^2 + 5
√
x− 1
dx
6.
∫
tan
(
x
2
)
dx
7.
∫
xcsc^2 (x^2 )dx
8.
∫
sinx
cos^3 x
dx
9.
∫
1
x^2 + 2 x+ 10
dx
10.
∫
1
x^2
sec^2
(
1
x
)
dx
11.
∫
(e^2 x)(e^4 x)dx
12.
∫
1
xlnx
dx