Power Rule Integration Calculator

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

This Power Rule Integration Calculator integrates a variable raised to a power, and shows the working line by line. The exponent goes up by one and the term is divided by that new exponent, which is exactly the power rule for differentiation run backwards. An exponent of negative one is the one case this cannot handle, because dividing by zero is not allowed. Following the same pattern every time keeps the raise and the divide in order.

Step-by-step method

  1. Set up the problem.
  2. Write the power rule for integration.
  3. Raise the exponent by one and divide by the new exponent.
  4. Simplify and add the constant of integration C.

Formula:

\(\int x^{n}\, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1\)
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the power rule for integration.
  3. Raise the exponent by one and divide by the new exponent.
  4. Simplify and add the constant of integration C.

Step 1

Step 1 - Identify the exponent n (and any constant multiple).

In this problem: The exponent is \(n = 3\).

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

Step 2

Step 2 - Write the power rule for integration.

In this problem: Raise the exponent by one, then divide by the new exponent.

\(\int x^{n}\, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1\)

Step 3

Step 3 - Raise the exponent by one and divide by the new exponent.

In this problem: The new exponent is \(3 + 1 = 4\).

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

Step 4

Step 4 - Simplify and add the constant of integration C.

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

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

Final Answer

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

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the power rule for integration.
  3. Raise the exponent by one and divide by the new exponent.
  4. Simplify and add the constant of integration C.

Step 1

Step 1 - Identify the exponent n (and any constant multiple).

In this problem: The exponent is \(n = 2\). The constant multiple \(5\) stays in front.

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

Step 2

Step 2 - Write the power rule for integration.

In this problem: Raise the exponent by one, then divide by the new exponent.

\(\int x^{n}\, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1\)

Step 3

Step 3 - Raise the exponent by one and divide by the new exponent.

In this problem: The new exponent is \(2 + 1 = 3\).

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

Step 4

Step 4 - Simplify and add the constant of integration C.

In this problem: The simplified antiderivative is \(\frac{5 x^{3}}{3}\).

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

Final Answer

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

Example Problems

Click one to load it into the box and solve it.