5.2. PARTIAL FRACTIONS 173
Z
dx
x
3
1
D
1
3
Z
1
x 1
x C2
x
2
C x C1
dx
D
1
3
Z
1
x 1
1
2
2x C4
x
2
C x C1
dx
D
1
3
Z
1
x 1
1
2
2x C1 C 3
x
2
C x C1
dx
D
1
3
Z
1
x 1
1
2
2x C1
x
2
C x C1
1
2
3
x
2
C x C1
dx
e middle term is a u-substitution that leads to a log.
D
1
3
ln jx 1j
1
6
ln jx
2
C x C1j
1
2
Z
dx
x
2
C x C1
D
1
3
ln jx 1j
1
6
ln jx
2
C x C1j
1
2
Z
dx
.x C1=2/
2
C 3=4
D
1
3
ln jx 1j
1
6
ln jx
2
C x C1j
1
2
Z
dx
3
4
4
3
.x C1=2/
2
C 1
D
1
3
ln jx 1j
1
6
ln jx
2
C x C1j
1
2
4
3
Z
dx
4
3
.x C1=2/
2
C 1
D
1
3
ln jx 1j
1
6
ln jx
2
C x C1j
2
3
Z
dx
2
p
3
.x C1=2/
2
C 1
Let u D
2
p
3
.x C
1
2
/ and so du D
2
p
3
dx
D
1
3
ln jx 1j
1
6
ln jx
2
C x C1j
2
3
p
3
2
Z
du
u
2
C 1
D
1
3
ln jx 1j
1
6
ln jx
2
C x C1j
1
p
3
tan
1
.u/ CC
174 5. METHODS OF INTEGRATION II
D
1
3
ln jx 1j
1
6
ln jx
2
C x C1j
1
p
3
tan
1
2
p
3
x C
1
2
C C
See! Easy!
˙
One of the fairly obvious conclusions that follows from this section is that you need to not only
know algebra, you need to be a master of it to mess with partial fractions in all but its simplest
form. Some of the homework problems develop general formulas for this sort of super messy
inverse tangent integral. Like Knowledge Box 5.4, but with more parameters.
PROBLEMS
Problem 5.26 Perform the following integrals.
1.
Z
dx
x
2
x
2.
Z
dx
x
2
6x C8
3.
Z
dx
x
2
C 4x C 3
4.
Z
dx
6x
2
5x C1
5.
Z
dx
x
2
x 1
6.
Z
dx
x
2
11x C30
Problem 5.27 Perform each of the following integrals. Partial fractions are not needed.
1.
Z
1
x
2
C 2x C2
dx
2.
Z
1
x
2
C 3x C4
dx
3.
Z
1
x
2
C x C2
dx
4.
Z
x C1
x
2
C 2x C2
dx
5.
Z
2x C2
x
2
C 3x C4
dx
6.
Z
x 1
x
2
C x C2
dx
Problem 5.28 Perform each of the following integrals.
1.
Z
dx
x
3
C 6x
2
C 11x C6
2.
Z
dx
x
4
16
3.
Z
dx
x
3
5x
2
2x C24
4.
Z
x
2
C x C1
x
3
6x
2
C 11x C6
dx
5.
Z
x
2
C 1
x
3
C 4x
dx
6.
Z
3x 2
x
3
C 2x
2
5x 6
dx
5.2. PARTIAL FRACTIONS 175
Problem 5.29 For each of the following, compute the partial fractions form for decomposing
the integral, but do not solve for the partial fractions coefficients or perform the integral.
1.
Z
dx
x
5
C 4x
3
C x
2.
Z
dx
x
4
C 4x
3
C 7x
2
C 6x C3
3.
Z
dx
x
5
C x
3
4.
Z
dx
x
6
5x
5
C6x
4
x
3
C5x
2
5x C1
5.
Z
dx
x
6
3x
5
C 3x
4
C x
3
6.
Z
dx
x
6
C2x
5
C5x
3
C6x
2
C3x C2
Problem 5.30 Compute
Z
dx
ax
2
C bx Cc
ere are three cases each of which requires its own method of integration.
Problem 5.31 Compute
Z
dx
x
3
a
3
where a is a positive constant.
Problem 5.32 Compute
Z
dx
x
4
a
4
where a is a positive constant.
Problem 5.33
Perform the integrals in Problem
5.29.
Problem 5.34 Compute
Z
x
5
x
2
C x C1
dx
Problem 5.35 Compute
Z
dx
.x a/.x b/.x c/
dx
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