Sum Rule Integration Calculator

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

This Sum Rule Integration Calculator helps you integrate sums such as x^2 + x or x^3 + 2x + 1. The sum rule says the integral of a sum is the sum of the integrals, so the integral splits across the addition signs.

Step-by-step method

  1. Set up the terms in the sum.
  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:

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

Example 1:

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

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 - 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 - 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 - 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 - 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:

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

Example 2:

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

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 - 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 - 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 - 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 - 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:

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