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Calculus II · Techniques of Integration

Partial Fractions — Practice Problems and Worked Solutions

Split rational functions into simpler logarithmic pieces.

The formulas you need here

Partial fractions

\frac{P ( x )}{( x - a ) ( x - b )} = \frac{A}{x - a} + \frac{B}{x - b}

What it says: Split one hard fraction into a sum of easy ones, each integrating to a logarithm.
When to reach for it: A rational function whose denominator factors and whose top has the lower degree.
Watch out: If the numerator degree is not lower, do polynomial division first. A repeated factor needs both A/(x−a) and B/(x−a)².

\frac{1}{x ^{2} - 1} = \frac{1/2}{x - 1} - \frac{1/2}{x + 1}

Each piece integrates to a log, so the answer is a difference of logs.

16 worked problems

Every step is generated and checked by a computer algebra system.

1.
Integrate using partial fractions.
$\int \frac{1}{( x + 1 ) ( x + 2 )} d x$
Answer: $lo g (x + 1) - lo g (x + 2) + C$
Step-by-step solution
1. $\frac{1}{( x + 1 ) ( x + 2 )} = - \frac{1}{x + 2} + \frac{1}{x + 1}$
2. $\int - \frac{1}{x + 2} + \frac{1}{x + 1} d x = lo g (x + 1) - lo g (x + 2) + C$
2.
Integrate using partial fractions.
$\int \frac{2 x + 3}{( x + 1 ) ( x + 3 )} d x$
Answer: $\frac{l o g ( x + 1 )}{2} + \frac{3 l o g ( x + 3 )}{2} + C$
Step-by-step solution
1. $\frac{2 x + 3}{( x + 1 ) ( x + 3 )} = \frac{3}{2 ( x + 3 )} + \frac{1}{2 ( x + 1 )}$
2. $\int \frac{3}{2 ( x + 3 )} + \frac{1}{2 ( x + 1 )} d x = \frac{lo g ( x + 1 )}{2} + \frac{3 lo g ( x + 3 )}{2} + C$
3.
Integrate using partial fractions.
$\int \frac{x}{x ^{2} - 1} d x$
Answer: $\frac{l o g ( x ^{2} - 1 )}{2} + C$
Step-by-step solution
1. $\frac{x}{x ^{2} - 1} = \frac{1}{2 ( x + 1 )} + \frac{1}{2 ( x - 1 )}$
2. $\int \frac{1}{2 ( x + 1 )} + \frac{1}{2 ( x - 1 )} d x = \frac{lo g ( x ^{2} - 1 )}{2} + C$
4.
Integrate using partial fractions.
$\int \frac{3 x + 5}{( x - 1 ) ( x + 2 )} d x$
Answer: $\frac{8 l o g ( x - 1 )}{3} + \frac{l o g ( x + 2 )}{3} + C$
Step-by-step solution
1. $\frac{3 x + 5}{( x - 1 ) ( x + 2 )} = \frac{1}{3 ( x + 2 )} + \frac{8}{3 ( x - 1 )}$
2. $\int \frac{1}{3 ( x + 2 )} + \frac{8}{3 ( x - 1 )} d x = \frac{8 lo g ( x - 1 )}{3} + \frac{lo g ( x + 2 )}{3} + C$
5.
Integrate using partial fractions.
$\int \frac{x ^{2} + 1}{x ( x + 1 )} d x$
Answer: $x + lo g (x) - 2 lo g (x + 1) + C$
Step-by-step solution
1. $\frac{x ^{2} + 1}{x ( x + 1 )} = 1 - \frac{2}{x + 1} + \frac{1}{x}$
2. $\int 1 - \frac{2}{x + 1} + \frac{1}{x} d x = x + lo g (x) - 2 lo g (x + 1) + C$
6.
Integrate using partial fractions.
$\int \frac{2 x ^{2} + 3 x + 1}{x ( x + 1 ) ( x + 2 )} d x$
Answer: $\frac{l o g ( x )}{2} + \frac{3 l o g ( x + 2 )}{2} + C$
Step-by-step solution
1. $\frac{2 x ^{2} + 3 x + 1}{x ( x + 1 ) ( x + 2 )} = \frac{3}{2 ( x + 2 )} + \frac{1}{2 x}$
2. $\int \frac{3}{2 ( x + 2 )} + \frac{1}{2 x} d x = \frac{lo g ( x )}{2} + \frac{3 lo g ( x + 2 )}{2} + C$
7.
Integrate using partial fractions.
$\int \frac{1}{( x + 2 ) ( x + 3 )} d x$
Answer: $lo g (x + 2) - lo g (x + 3) + C$
Step-by-step solution
1. $\frac{1}{( x + 2 ) ( x + 3 )} = - \frac{1}{x + 3} + \frac{1}{x + 2}$
2. $\int - \frac{1}{x + 3} + \frac{1}{x + 2} d x = lo g (x + 2) - lo g (x + 3) + C$
8.
Integrate using partial fractions.
$\int \frac{1}{( x - 1 ) ( x + 1 )} d x$
Answer: $\frac{l o g ( x - 1 )}{2} - \frac{l o g ( x + 1 )}{2} + C$
Step-by-step solution
1. $\frac{1}{( x - 1 ) ( x + 1 )} = - \frac{1}{2 ( x + 1 )} + \frac{1}{2 ( x - 1 )}$
2. $\int - \frac{1}{2 ( x + 1 )} + \frac{1}{2 ( x - 1 )} d x = \frac{lo g ( x - 1 )}{2} - \frac{lo g ( x + 1 )}{2} + C$
9.
Integrate using partial fractions.
$\int \frac{x + 4}{( x + 1 ) ( x + 2 )} d x$
Answer: $3 lo g (x + 1) - 2 lo g (x + 2) + C$
Step-by-step solution
1. $\frac{x + 4}{( x + 1 ) ( x + 2 )} = - \frac{2}{x + 2} + \frac{3}{x + 1}$
2. $\int - \frac{2}{x + 2} + \frac{3}{x + 1} d x = 3 lo g (x + 1) - 2 lo g (x + 2) + C$
10.
Integrate using partial fractions.
$\int \frac{x}{x ^{2} - 4} d x$
Answer: $\frac{l o g ( x ^{2} - 4 )}{2} + C$
Step-by-step solution
1. $\frac{x}{x ^{2} - 4} = \frac{1}{2 ( x + 2 )} + \frac{1}{2 ( x - 2 )}$
2. $\int \frac{1}{2 ( x + 2 )} + \frac{1}{2 ( x - 2 )} d x = \frac{lo g ( x ^{2} - 4 )}{2} + C$
11.
Integrate using partial fractions.
$\int \frac{2 x + 1}{( x - 2 ) ( x + 3 )} d x$
Answer: $lo g (x^{2} + x - 6) + C$
Step-by-step solution
1. $\frac{2 x + 1}{( x - 2 ) ( x + 3 )} = \frac{1}{x + 3} + \frac{1}{x - 2}$
2. $\int \frac{1}{x + 3} + \frac{1}{x - 2} d x = lo g (x^{2} + x - 6) + C$
12.
Integrate using partial fractions.
$\int \frac{1}{x ( x + 4 )} d x$
Answer: $\frac{l o g ( x )}{4} - \frac{l o g ( x + 4 )}{4} + C$
Step-by-step solution
1. $\frac{1}{x ( x + 4 )} = - \frac{1}{4 ( x + 4 )} + \frac{1}{4 x}$
2. $\int - \frac{1}{4 ( x + 4 )} + \frac{1}{4 x} d x = \frac{lo g ( x )}{4} - \frac{lo g ( x + 4 )}{4} + C$
13.
Integrate using partial fractions.
$\int \frac{3 x - 1}{( x - 1 ) ( x + 2 )} d x$
Answer: $\frac{2 l o g ( x - 1 )}{3} + \frac{7 l o g ( x + 2 )}{3} + C$
Step-by-step solution
1. $\frac{3 x - 1}{( x - 1 ) ( x + 2 )} = \frac{7}{3 ( x + 2 )} + \frac{2}{3 ( x - 1 )}$
2. $\int \frac{7}{3 ( x + 2 )} + \frac{2}{3 ( x - 1 )} d x = \frac{2 lo g ( x - 1 )}{3} + \frac{7 lo g ( x + 2 )}{3} + C$
14.
Integrate using partial fractions.
$\int \frac{x ^{2} + 2}{x ( x + 2 )} d x$
Answer: $x + lo g (x) - 3 lo g (x + 2) + C$
Step-by-step solution
1. $\frac{x ^{2} + 2}{x ( x + 2 )} = 1 - \frac{3}{x + 2} + \frac{1}{x}$
2. $\int 1 - \frac{3}{x + 2} + \frac{1}{x} d x = x + lo g (x) - 3 lo g (x + 2) + C$
15.
Integrate using partial fractions.
$\int \frac{5}{( x + 1 ) ( x + 6 )} d x$
Answer: $lo g (x + 1) - lo g (x + 6) + C$
Step-by-step solution
1. $\frac{5}{( x + 1 ) ( x + 6 )} = - \frac{1}{x + 6} + \frac{1}{x + 1}$
2. $\int - \frac{1}{x + 6} + \frac{1}{x + 1} d x = lo g (x + 1) - lo g (x + 6) + C$
16.
Integrate using partial fractions.
$\int \frac{x + 7}{( x - 3 ) ( x + 2 )} d x$
Answer: $2 lo g (x - 3) - lo g (x + 2) + C$
Step-by-step solution
1. $\frac{x + 7}{( x - 3 ) ( x + 2 )} = - \frac{1}{x + 2} + \frac{2}{x - 3}$
2. $\int - \frac{1}{x + 2} + \frac{2}{x - 3} d x = 2 lo g (x - 3) - lo g (x + 2) + C$

The formulas you need here

Partial fractions

16 worked problems

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