Partial Fractions Integration Notes

Published on: September 29, 2026

This Partial Fractions Integration Calculator integrates a fraction of polynomials by splitting it into simpler pieces, and shows the working line by line. The denominator is factored and the fraction is rewritten as a sum whose parts each integrate to a logarithm or a power. Splitting first is what turns an impossible integral into routine ones. Following the same pattern every time keeps the split and the integration as separate moves.

Step-by-step method

  1. Set up the problem.
  2. Decompose the fraction into partial fractions.
  3. Integrate each partial fraction separately.
  4. Combine the results and add the constant of integration C.

Formula:

\(\frac{P\left(x\right)}{Q\left(x\right)} = \frac{A}{x - a} + \frac{B}{x - b} + \cdots\)
See Example 1 Hide Example 1

Problem

\(\int \frac{3 x + 5}{x^{2} + 3 x + 2}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Decompose the fraction into partial fractions.
  3. Integrate each partial fraction separately.
  4. Combine the results and add the constant of integration C.

Step 1

Step 1 - Set up the rational function and check the degrees.

In this problem: The numerator is \(3 x + 5\) and the denominator is \(x^{2} + 3 x + 2\).

\(\int \frac{3 x + 5}{x^{2} + 3 x + 2}\, dx\)

Step 2

Step 2 - Decompose the fraction into partial fractions.

In this problem: Splitting the fraction gives \(\frac{1}{x + 2} + \frac{2}{x + 1}\).

\(\frac{3 x + 5}{x^{2} + 3 x + 2} = \frac{1}{x + 2} + \frac{2}{x + 1}\)

Step 3

Step 3 - Integrate each partial fraction separately.

In this problem: Each simple fraction integrates to a logarithm or a power.

\(\int \frac{1}{x + 2}\, dx + \int \frac{2}{x + 1}\, dx = \ln\left(x + 2\right) + 2 \ln\left(x + 1\right) + C\)

Step 4

Step 4 - Combine the results and add the constant of integration C.

In this problem: The combined antiderivative is \(2 \ln\left(x + 1\right) + \ln\left(x + 2\right)\).

\(\int \frac{3 x + 5}{x^{2} + 3 x + 2}\, dx = 2 \ln\left(x + 1\right) + \ln\left(x + 2\right) + C\)

Final Answer

\(\int \frac{3 x + 5}{x^{2} + 3 x + 2}\, dx = 2 \ln\left(x + 1\right) + \ln\left(x + 2\right) + C\)
See Example 2 Hide Example 2

Problem

\(\int \frac{1}{x^{2} - 1}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Decompose the fraction into partial fractions.
  3. Integrate each partial fraction separately.
  4. Combine the results and add the constant of integration C.

Step 1

Step 1 - Set up the rational function and check the degrees.

In this problem: The numerator is \(1\) and the denominator is \(x^{2} - 1\).

\(\int \frac{1}{x^{2} - 1}\, dx\)

Step 2

Step 2 - Decompose the fraction into partial fractions.

In this problem: Splitting the fraction gives \(- \frac{1}{2 \left(x + 1\right)} + \frac{1}{2 \left(x - 1\right)}\).

\(\frac{1}{x^{2} - 1} = \frac{1}{2 \left(x - 1\right)} - \frac{1}{2 \left(x + 1\right)}\)

Step 3

Step 3 - Integrate each partial fraction separately.

In this problem: Each simple fraction integrates to a logarithm or a power.

\(\int \frac{1}{2 \left(x - 1\right)}\, dx - \int \frac{1}{2 \left(x + 1\right)}\, dx = \frac{\ln\left(2 x - 2\right)}{2} - \frac{\ln\left(2 x + 2\right)}{2} + C\)

Step 4

Step 4 - Combine the results and add the constant of integration C.

In this problem: The combined antiderivative is \(\frac{\ln\left(2 x - 2\right)}{2} - \frac{\ln\left(2 x + 2\right)}{2}\).

\(\int \frac{1}{x^{2} - 1}\, dx = \frac{\ln\left(2 x - 2\right)}{2} - \frac{\ln\left(2 x + 2\right)}{2} + C\)

Final Answer

\(\int \frac{1}{x^{2} - 1}\, dx = \frac{\ln\left(2 x - 2\right)}{2} - \frac{\ln\left(2 x + 2\right)}{2} + C\)
See Example 3 Hide Example 3

Problem

\(\int \frac{2 x + 3}{x^{2} + 3 x + 2}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Decompose the fraction into partial fractions.
  3. Integrate each partial fraction separately.
  4. Combine the results and add the constant of integration C.

Step 1

Step 1 - Set up the rational function and check the degrees.

In this problem: The numerator is \(2 x + 3\) and the denominator is \(x^{2} + 3 x + 2\).

\(\int \frac{2 x + 3}{x^{2} + 3 x + 2}\, dx\)

Step 2

Step 2 - Decompose the fraction into partial fractions.

In this problem: Splitting the fraction gives \(\frac{1}{x + 2} + \frac{1}{x + 1}\).

\(\frac{2 x + 3}{x^{2} + 3 x + 2} = \frac{1}{x + 1} + \frac{1}{x + 2}\)

Step 3

Step 3 - Integrate each partial fraction separately.

In this problem: Each simple fraction integrates to a logarithm or a power.

\(\int \frac{1}{x + 1}\, dx + \int \frac{1}{x + 2}\, dx = \ln\left(x + 1\right) + \ln\left(x + 2\right) + C\)

Step 4

Step 4 - Combine the results and add the constant of integration C.

In this problem: The combined antiderivative is \(\ln\left(x + 1\right) + \ln\left(x + 2\right)\).

\(\int \frac{2 x + 3}{x^{2} + 3 x + 2}\, dx = \ln\left(x + 1\right) + \ln\left(x + 2\right) + C\)

Final Answer

\(\int \frac{2 x + 3}{x^{2} + 3 x + 2}\, dx = \ln\left(x + 1\right) + \ln\left(x + 2\right) + C\)
See Example 4 Hide Example 4

Problem

\(\int \frac{1}{x^{2} - 9}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Decompose the fraction into partial fractions.
  3. Integrate each partial fraction separately.
  4. Combine the results and add the constant of integration C.

Step 1

Step 1 - Set up the rational function and check the degrees.

In this problem: The numerator is \(1\) and the denominator is \(x^{2} - 9\).

\(\int \frac{1}{x^{2} - 9}\, dx\)

Step 2

Step 2 - Decompose the fraction into partial fractions.

In this problem: Splitting the fraction gives \(- \frac{1}{6 \left(x + 3\right)} + \frac{1}{6 \left(x - 3\right)}\).

\(\frac{1}{x^{2} - 9} = -\frac{1}{6 \left(x + 3\right)} + \frac{1}{6 \left(x - 3\right)}\)

Step 3

Step 3 - Integrate each partial fraction separately.

In this problem: Each simple fraction integrates to a logarithm or a power.

\(-\int \frac{1}{6 \left(x + 3\right)}\, dx + \int \frac{1}{6 \left(x - 3\right)}\, dx = -\frac{\ln\left(6 x + 18\right)}{6} + \frac{\ln\left(6 x - 18\right)}{6} + C\)

Step 4

Step 4 - Combine the results and add the constant of integration C.

In this problem: The combined antiderivative is \(\frac{\ln\left(6 x - 18\right)}{6} - \frac{\ln\left(6 x + 18\right)}{6}\).

\(\int \frac{1}{x^{2} - 9}\, dx = \frac{\ln\left(6 x - 18\right)}{6} - \frac{\ln\left(6 x + 18\right)}{6} + C\)

Final Answer

\(\int \frac{1}{x^{2} - 9}\, dx = \frac{\ln\left(6 x - 18\right)}{6} - \frac{\ln\left(6 x + 18\right)}{6} + C\)