Sum Rule Integration Notes

Published on: September 29, 2026

This Sum Rule Integration Calculator integrates two or more terms added together, and shows the working line by line. The integral splits across the plus signs, so each term is integrated on its own and the results are added back. One constant of integration covers the whole answer rather than one per term. Following the same pattern every time keeps each term's integral traceable to its own term.

Step-by-step method

  1. Set up the problem.
  2. Write the sum rule for integration.
  3. Split the integral across the addition 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

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the sum rule for integration.
  3. Split the integral across the addition 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 sum.

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

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

Step 2

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

Step 3 - Split the integral across the addition signs.

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

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

Step 4

Step 4 - Integrate each separated term.

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

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

Step 5

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

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

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

Final Answer

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

Problem

\(\int x^{3} + 2 x + 1\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the sum rule for integration.
  3. Split the integral across the addition 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 sum.

In this problem: We are integrating a sum. The separate terms are \(x^{3}\), \(2 x\), \(1\).

\(\int x^{3} + 2 x + 1\, dx\)

Step 2

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

Step 3 - Split the integral across the addition signs.

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

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

Step 4

Step 4 - Integrate each separated term.

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

\(\int x^{3}\, dx + \int 2 x\, dx + \int 1\, dx = \frac{x^{4}}{4} + x^{2} + x + C\)

Step 5

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

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

\(\int \left(x^{3} + 2 x + 1\right)\, dx = \frac{x^{4}}{4} + x^{2} + x + C\)

Final Answer

\(\int x^{3} + 2 x + 1\, dx = \frac{x^{4}}{4} + x^{2} + x + C\)
See Example 3 Hide Example 3

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the sum rule for integration.
  3. Split the integral across the addition 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 sum.

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

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

Step 2

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

Step 3 - Split the integral across the addition signs.

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

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

Step 4

Step 4 - Integrate each separated term.

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

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

Step 5

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

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

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

Final Answer

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

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the sum rule for integration.
  3. Split the integral across the addition 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 sum.

In this problem: We are integrating a sum. The separate terms are \(x^{2}\), \(4 x\), \(3\).

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

Step 2

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

Step 3 - Split the integral across the addition signs.

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

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

Step 4

Step 4 - Integrate each separated term.

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

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

Step 5

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

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

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

Final Answer

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