370 STEP 4. Review the Knowledge You Need to Score High
- Integrate by parts, usingu=lnx,
dv=x^3 dx,du=1
xdx,v=
x^4
4. Then
∫
x^3 lnxdx=
x^4
4
lnx−
∫
x^4
4·
1
x
dx=x^4
4
lnx−1
4
∫
x^3 dx=
x^4
4
lnx−
x^4
16.
Consider the limits of integration,
∫e1x^3 lnxdx=
x^4
4
lnx−
x^4
16∣
∣∣
∣e1=
(
e^4
4
lne−
e^4
16)
−(
14
4
ln 1−14
16
)
=3 e^4 + 1
16≈ 10 .300.
- Use partial fraction decomposition.
5
x^2 −x− 6
=
5
(x−3)(x+2)=
A
x− 3+
B
x+ 2.
Solving the system{
A+B= 0
2 A− 3 B= 5givesA=1,B=−1, and
∫ 105
x^2 −x− 6
dx=∫ 101
x− 3
dx+
∫ 10− 1
x+ 2
dx. Then ln|x− 3 ||^10−ln|x+ 2 ||^10 =2ln2−2ln3≈− 0 .811.- limx→ 1
lnx
x^2 − 1
=limx→ 1
1 /x
2 x
=limx→ 1
1
2 x^2