Indefinite Integral Calculator

Published on: May 4, 2025
Final Answer: Free • Full Steps: Plus

This Indefinite Integral Calculator finds the antiderivative of a function and shows the working line by line. It first works out which rule of integration the function needs, since a power, a reciprocal, an exponential and a quotient are each undone differently, and only then does it apply that rule. The constant of integration is added at the end, because any constant vanishes when differentiated. Following the same pattern every time keeps the choice of rule visible.

Step-by-step method

  1. Set up the problem.
  2. Identify the integration rule that applies.
  3. Work through the steps of that rule.
  4. Simplify and add the constant of integration C.

Formula:

\(\int f\left(x\right) \, dx = F\left(x\right) + C\)
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Identify the integration rule that applies.
  3. Work through the steps of that rule.
  4. Simplify 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 \(x^{5}\), \(3 x^{2}\).

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

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.

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

Step 4

Step 4 - Integrate each separated term.

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

\(\int x^{5}\, dx - \int 3 x^{2}\, dx = \frac{x^{6}}{6} - x^{3} + C\)

Step 5

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

In this problem: The combined antiderivative is \(\frac{x^{6}}{6} - x^{3}\).

\(\int \left(x^{5} - 3 x^{2}\right)\, dx = \frac{x^{6}}{6} - x^{3} + C\)

Final Answer

\(\int x^{5} - 3 x^{2}\, dx = \frac{x^{6}}{6} - x^{3} + C\)
See Example 2 Hide Example 2

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Identify the integration rule that applies.
  3. Work through the steps of that rule.
  4. Simplify and add the constant of integration C.

Step 1

Step 1 - Set up the problem.

In this problem: The trigonometric form is \(\sin{\left(x \right)}\).

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

Step 2

Step 2 - Write the integration rule for that form.

In this problem: Each standard trigonometric form has a known antiderivative.

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

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front, and add C.

In this problem: Replace the trigonometric form with its antiderivative and simplify.

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

Final Answer

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

Example Problems

Click one to load it into the box and solve it.