Difference Rule Integration Notes

Published on: September 29, 2026

This Difference Rule Integration Calculator integrates terms that are subtracted from one another, and shows the working line by line. The integral splits across the minus signs just as it does across plus signs, with each term handled separately and the signs left as they were. A single constant of integration covers the whole answer. Following the same pattern every time keeps each sign attached to its own term.

Step-by-step method

  1. Set up the problem.
  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:

\(\int \left(f\left(x\right) - g\left(x\right)\right)\, dx = \int f\left(x\right)\, dx - \int g\left(x\right)\, dx\)
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  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.

Step 1

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

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

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

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

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

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

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  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.

Step 1

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

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

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

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

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

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

Problem

\(\int x^{4} - x^{2}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  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.

Step 1

Step 1 - Set up the terms in the difference.

In this problem: We are integrating a difference. The separated terms are \(x^{4}\), \(x^{2}\).

\(\int x^{4} - x^{2}\, dx\)

Step 2

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

Step 3 - Split the integral across the subtraction signs.

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

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

Step 4

Step 4 - Integrate each separated term.

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

\(\int x^{4}\, dx - \int x^{2}\, dx = \frac{x^{5}}{5} - \frac{x^{3}}{3} + C\)

Step 5

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

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

\(\int \left(x^{4} - x^{2}\right)\, dx = \frac{x^{5}}{5} - \frac{x^{3}}{3} + C\)

Final Answer

\(\int x^{4} - x^{2}\, dx = \frac{x^{5}}{5} - \frac{x^{3}}{3} + C\)
See Example 4 Hide Example 4

Problem

\(\int 6 x^{2} - 4 x\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  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.

Step 1

Step 1 - Set up the terms in the difference.

In this problem: We are integrating a difference. The separated terms are \(6 x^{2}\), \(4 x\).

\(\int 6 x^{2} - 4 x\, dx\)

Step 2

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

Step 3 - Split the integral across the subtraction signs.

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

\(\int \left(6 x^{2} - 4 x\right)\, dx = \int 6 x^{2}\, dx - \int 4 x\, dx\)

Step 4

Step 4 - Integrate each separated term.

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

\(\int 6 x^{2}\, dx - \int 4 x\, dx = 2 x^{3} - 2 x^{2} + C\)

Step 5

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

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

\(\int \left(6 x^{2} - 4 x\right)\, dx = 2 x^{3} - 2 x^{2} + C\)

Final Answer

\(\int 6 x^{2} - 4 x\, dx = 2 x^{3} - 2 x^{2} + C\)