Calculators / Calculus 1 / Integral Calculator

Integral Calculator

Published on: May 4, 2025

This Integral Calculator helps you compute indefinite integrals and shows each step clearly. It can be used to integrate common functions by applying standard integration rules, simplifying expressions, and identifying the correct variable of integration. This makes it useful for checking answers, understanding how integration works, and practising calculus step by step.

Step-by-step method

  1. Write the problem.
  2. Split the integral (if it's a sum or difference).
  3. Integrate each piece using the right formula.
  4. Combine the results, and simplify only if needed.
  5. Add + C.

Formula:

∫ c dx = cx
∫ x dx =
x^2
2
∫ xn dx =
xn+1
n+1
∫ k·f(x) dx = k·∫ f(x) dx
∫ (u ± v) dx = ∫ u dx ± ∫ v dx
∫ sin(x) dx = −cos(x)
∫ cos(x) dx = sin(x)
∫ exp(x) dx = exp(x)
∫ 1/x dx = ln(x)

Example 1: sin(x)+x^2, x

Step 1 - Write the problem.

In this problem: Integrate with respect to x.

∫x2 + sin(x)dx

Step 2A - Show the formula.

In this problem: Because this is a sum/difference, we can split it using this rule:

∫ (u ± v) dx = ∫ u dx ± ∫ v dx

Step 2B - Split the integral.

In this problem: Split the sum/difference into separate integrals.

∫(x2 + sin(x))dx=∫x2dx + ∫sin(x)dx

Step 3A - Focus on this integral.

In this problem: Work on one piece at a time.

∫x2dx

Step 3B - Show the formula.

In this problem: Use the matching integration formula.

∫ xn dx =
xn+1
n+1

Step 3C - Apply the formula.

In this problem: Compute the antiderivative for this piece.

∫x2dx=
x3
3

Step 3D - Focus on this integral.

In this problem: Work on one piece at a time.

∫sin(x)dx

Step 3E - Show the formula.

In this problem: Use the matching integration formula.

∫ sin(x) dx = −cos(x)

Step 3F - Apply the formula.

In this problem: Compute the antiderivative for this piece.

∫sin(x)dx=−cos(x)

Step 3G - Combine the results.

In this problem: Add the pieces back together.

∫x2 + sin(x)dx=
x3
3
− cos(x)

Final answer: ∫ x^2 + sin(x) dx = x^3/3 - cos(x) + C

Example 2: x^2+3x, x

Step 1 - Write the problem.

In this problem: Integrate with respect to x.

∫x2 + 3xdx

Step 2A - Show the formula.

In this problem: Because this is a sum/difference, we can split it using this rule:

∫ (u ± v) dx = ∫ u dx ± ∫ v dx

Step 2B - Split the integral.

In this problem: Split the sum/difference into separate integrals.

∫(x2 + 3x)dx=∫x2dx + ∫3xdx

Step 3A - Focus on this integral.

In this problem: Work on one piece at a time.

∫x2dx

Step 3B - Show the formula.

In this problem: Use the matching integration formula.

∫ xn dx =
xn+1
n+1

Step 3C - Apply the formula.

In this problem: Compute the antiderivative for this piece.

∫x2dx=
x3
3

Step 3D - Focus on this integral.

In this problem: Work on one piece at a time.

∫3xdx

Step 3E - Show the formula.

In this problem: Use the matching integration formula.

∫ k·f(x) dx = k·∫ f(x) dx

Step 3F - Apply the formula.

In this problem: Compute the antiderivative for this piece.

∫3xdx=
3x2
2

Step 3G - Combine the results.

In this problem: Add the pieces back together.

∫x2 + 3xdx=
x2(2x + 9)
6

Final answer: ∫ x^2 + 3x dx = x^2(2x + 9)/6 + C