Indefinite Integral Calculator

Published on: May 4, 2025
Final Answer: Free • Full Steps: Plus

This Indefinite Integral Calculator integrates a function step by step. It first identifies which integration rule fits your input - constant, power, constant multiple, sum, difference, exponential, reciprocal, or partial fractions - and then works through that rule's own method, the same way the dedicated rule calculators on this site do. Every answer includes the constant of integration C.

Step-by-step method

  1. Set up the integral.
  2. Identify the integration rule that applies.
  3. Work through the steps of that rule.
  4. Simplify and add the constant of integration C.

Formula:

\(\int f\left(x\right) \, dx = F\left(x\right) + C\)

Example 1: solved with the sum rule for integration.

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

Step 1 - Set up the terms in the sum.

In this problem: We are integrating a sum. The separate terms are \(x^{3}\), \(2 x\).

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

Step 2 - Write the sum rule for integration.

In this problem: The integral of a sum is the sum of the integrals.

\(\int \left(f\left(x\right) + g\left(x\right)\right)\, dx = \int f\left(x\right)\, dx + \int g\left(x\right)\, dx\)

Step 3 - Split the integral across the addition signs.

In this problem: Apply the integral to each separated term.

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

Step 4 - Integrate each separated term.

In this problem: Integrate each term with the basic integration rules.

\(\int x^{3}\, dx + \int 2 x\, dx = \frac{x^{4}}{4} + x^{2} + C\)

Step 5 - Combine the results and add the constant of integration C.

In this problem: The combined antiderivative is \(\frac{x^{4}}{4} + x^{2}\).

\(\int \left(x^{3} + 2 x\right)\, dx = \frac{x^{4}}{4} + x^{2} + C\)

Final answer:

\(\int x^{3} + 2 x\, dx = \frac{x^{4}}{4} + x^{2} + C\)

Example 2: solved with the partial fractions.

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

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:

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