Constant Multiple Rule Integration Notes

Published on: September 29, 2026

This Constant Multiple Rule Integration Calculator integrates a function that has been scaled by a number, and shows the working line by line. The constant can be pulled outside the integral and multiplied back in afterwards, because scaling a function scales its area by the same factor. Only the function itself is actually integrated. Following the same pattern every time keeps the constant from being integrated along with the function.

Step-by-step method

  1. Set up the problem.
  2. Write the constant multiple rule for integration.
  3. Pull the constant outside the integral.
  4. Integrate the remaining function, multiply back, and add C.

Formula:

\(\int k \cdot f\left(x\right)\, dx = k\int f\left(x\right)\, dx\)
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the constant multiple rule for integration.
  3. Pull the constant outside the integral.
  4. Integrate the remaining function, multiply back, and add C.

Step 1

Step 1 - Identify the constant multiple k and the function f(x).

In this problem: The constant is \(k = 4\) and the function is \(x^{3}\).

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

Step 2

Step 2 - Write the constant multiple rule for integration.

In this problem: A constant factor can be moved outside the integral.

\(\int k \cdot f\left(x\right)\, dx = k\int f\left(x\right)\, dx\)

Step 3

Step 3 - Pull the constant outside the integral.

In this problem: Move \(4\) in front of the integral sign.

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

Step 4

Step 4 - Integrate the remaining function, multiply back, and add C.

In this problem: The integral of \(x^{3}\) is \(\frac{x^{4}}{4}\). Multiplying by \(4\) gives \(x^{4}\).

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

Final Answer

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

Problem

\(\int 2 \sin{\left(x \right)}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the constant multiple rule for integration.
  3. Pull the constant outside the integral.
  4. Integrate the remaining function, multiply back, and add C.

Step 1

Step 1 - Identify the constant multiple k and the function f(x).

In this problem: The constant is \(k = 2\) and the function is \(\sin{\left(x \right)}\).

\(\int 2 \sin{\left(x \right)}\, dx\)

Step 2

Step 2 - Write the constant multiple rule for integration.

In this problem: A constant factor can be moved outside the integral.

\(\int k \cdot f\left(x\right)\, dx = k\int f\left(x\right)\, dx\)

Step 3

Step 3 - Pull the constant outside the integral.

In this problem: Move \(2\) in front of the integral sign.

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

Step 4

Step 4 - Integrate the remaining function, multiply back, and add C.

In this problem: The integral of \(\sin{\left(x \right)}\) is \(- \cos{\left(x \right)}\). Multiplying by \(2\) gives \(- 2 \cos{\left(x \right)}\).

\(2\int \sin{\left(x \right)}\, dx = 2 \cdot \left(- \cos{\left(x \right)}\right) = - 2 \cos{\left(x \right)} + C\)

Final Answer

\(\int 2 \sin{\left(x \right)}\, dx = - 2 \cos{\left(x \right)} + C\)
See Example 3 Hide Example 3

Problem

\(\int 8 x^{3}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the constant multiple rule for integration.
  3. Pull the constant outside the integral.
  4. Integrate the remaining function, multiply back, and add C.

Step 1

Step 1 - Identify the constant multiple k and the function f(x).

In this problem: The constant is \(k = 8\) and the function is \(x^{3}\).

\(\int 8 x^{3}\, dx\)

Step 2

Step 2 - Write the constant multiple rule for integration.

In this problem: A constant factor can be moved outside the integral.

\(\int k \cdot f\left(x\right)\, dx = k\int f\left(x\right)\, dx\)

Step 3

Step 3 - Pull the constant outside the integral.

In this problem: Move \(8\) in front of the integral sign.

\(\int 8 x^{3}\, dx = 8\int x^{3}\, dx\)

Step 4

Step 4 - Integrate the remaining function, multiply back, and add C.

In this problem: The integral of \(x^{3}\) is \(\frac{x^{4}}{4}\). Multiplying by \(8\) gives \(2 x^{4}\).

\(8\int x^{3}\, dx = 8 \cdot \left(\frac{x^{4}}{4}\right) = 2 x^{4} + C\)

Final Answer

\(\int 8 x^{3}\, dx = 2 x^{4} + C\)
See Example 4 Hide Example 4

Problem

\(\int 3 \sin{\left(x \right)}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the constant multiple rule for integration.
  3. Pull the constant outside the integral.
  4. Integrate the remaining function, multiply back, and add C.

Step 1

Step 1 - Identify the constant multiple k and the function f(x).

In this problem: The constant is \(k = 3\) and the function is \(\sin{\left(x \right)}\).

\(\int 3 \sin{\left(x \right)}\, dx\)

Step 2

Step 2 - Write the constant multiple rule for integration.

In this problem: A constant factor can be moved outside the integral.

\(\int k \cdot f\left(x\right)\, dx = k\int f\left(x\right)\, dx\)

Step 3

Step 3 - Pull the constant outside the integral.

In this problem: Move \(3\) in front of the integral sign.

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

Step 4

Step 4 - Integrate the remaining function, multiply back, and add C.

In this problem: The integral of \(\sin{\left(x \right)}\) is \(- \cos{\left(x \right)}\). Multiplying by \(3\) gives \(- 3 \cos{\left(x \right)}\).

\(3\int \sin{\left(x \right)}\, dx = 3 \cdot \left(- \cos{\left(x \right)}\right) = - 3 \cos{\left(x \right)} + C\)

Final Answer

\(\int 3 \sin{\left(x \right)}\, dx = - 3 \cos{\left(x \right)} + C\)