Power Rule Integration Calculator

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

This Power Rule Integration Calculator helps you integrate powers of x such as x^3 or 5x^2. The power rule for integration raises the exponent by one and divides by the new exponent, then adds the constant of integration C.

Step-by-step method

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

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

Example 1:

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

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

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

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

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 - 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 - 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:

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

Example 2:

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

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 - 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 - 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 - 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:

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