Indefinite Integral Calculator

Published on: May 18, 2025

This Indefinite 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 indefinite integration works, and practising calculus step by step.

Step-by-step method

  1. Set up the indefinite integral.
  2. Pick the integration rule.
  3. Plug the problem into the rule.
  4. Integrate each piece.
  5. Combine results and add the constant of integration.

Formula:

∫(u ± v) dx = ∫u dx ± ∫v dx
∫k·u dx = k·∫u dx
∫ xn dx =
xn+1
n+1
+ C
∫
1
x
dx = ln|x| + C
∫ ex dx = ex + C
∫ sin(x) dx = −cos(x) + C
∫ cos(x) dx = sin(x) + C
∫ x dx =
x2
2
+ C
∫ c dx = c·x + C
Rule: computed symbolically

Example 1: x^2 + 1, x

Step 1 - Set up the indefinite integral.

In this problem: The integrand is x^2 + 1 and the variable is x.

∫x2 + 1 dx

Step 2 - Pick the rule.

In this problem: Use the Sum/Difference rule.

∫(u ± v) dx = ∫u dx ± ∫v dx

Step 3 - Plug into the rule.

In this problem: Split the integral across the terms.

∫x2 + 1 dx = ∫x2 dx + ∫1 dx

Step 4A - Set up this piece.

In this problem: Work one piece at a time.

∫x2 dx

Step 4B - Pick the rule.

In this problem: Use Power rule.

∫ xn dx =
xn+1
n+1
+ C

Step 4C - Apply the rule.

In this problem: Find an antiderivative.

∫ x2 dx =
1
3
x3

Step 4D - Set up this piece.

In this problem: Work one piece at a time.

∫1 dx

Step 4E - Pick the rule.

In this problem: Use Constant rule.

∫ c dx = c·x + C

Step 4F - Apply the rule.

In this problem: Find an antiderivative.

∫ 1 dx = x

Step 5 - Combine results and add C.

In this problem: Add the constant of integration because this is an indefinite integral.

1
3
x3 + x + C

Final answer: x^3/3 + x + C

Example 2: sin(x), x

Step 1 - Set up the indefinite integral.

In this problem: The integrand is sin(x) and the variable is x.

∫sin(x) dx

Step 2 - Pick the rule.

In this problem: Use the Trig rule (sin).

∫ sin(x) dx = −cos(x) + C

Step 3 - Plug into the rule.

In this problem: Use the identity for this integrand.

∫sin(x) dx

Step 4A - Set up this piece.

In this problem: Work one piece at a time.

∫sin(x) dx

Step 4B - Pick the rule.

In this problem: Use Trig rule (sin).

∫ sin(x) dx = −cos(x) + C

Step 4C - Apply the rule.

In this problem: Find an antiderivative.

∫ sin(x) dx = −cos(x)

Step 5 - Combine results and add C.

In this problem: Add the constant of integration because this is an indefinite integral.

−cos(x) + C

Final answer: -cos(x) + C