Difference Rule Integration Calculator

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

This Difference Rule Integration Calculator helps you integrate differences such as x^3 - x^2 or 5x^2 - 3x. The difference rule says the integral of a difference is the difference of the integrals, so the integral splits across the subtraction signs.

Step-by-step method

  1. Set up the terms in the difference.
  2. Write the difference rule for integration.
  3. Split the integral across the subtraction signs.
  4. Integrate each separated term.
  5. Combine the results and add the constant of integration C.

Formula:

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

Example 1:

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

Step 1 - Set up the terms in the difference.

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

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

Step 2 - Write the difference rule for integration.

In this problem: The integral of a difference is the difference 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 subtraction signs.

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

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

Step 4 - Integrate each separated term.

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

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

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

In this problem: The combined antiderivative is \(\frac{x^{4}}{4} - \frac{x^{3}}{3}\).

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

Final answer:

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

Example 2:

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

Step 1 - Set up the terms in the difference.

In this problem: We are integrating a difference. The separated terms are \(5 x^{2}\), \(3 x\).

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

Step 2 - Write the difference rule for integration.

In this problem: The integral of a difference is the difference 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 subtraction signs.

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

\(\int \left(5 x^{2} - 3 x\right)\, dx = \int 5 x^{2}\, dx - \int 3 x\, dx\)

Step 4 - Integrate each separated term.

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

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

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

In this problem: The combined antiderivative is \(\frac{5 x^{3}}{3} - \frac{3 x^{2}}{2}\).

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

Final answer:

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