Partial Fractions Integration Calculator

Published on: July 17, 2026
Final Answer: Free • Full Steps: Plus

This Partial Fractions Integration Calculator helps you integrate rational functions such as (3x + 5)/(x^2 + 3x + 2) or 1/(x^2 - 1). The fraction is first decomposed into simpler partial fractions, and each piece is then integrated separately, usually giving logarithms.

Step-by-step method

  1. Set up the rational function and check the degrees.
  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:

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

Example 1:

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

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 - 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 - 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 - 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:

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

Example 2:

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

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 - 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 - 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 - 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:

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