Calculators / Calculus 2 / Definite Integral Calculator

Definite Integral Calculator

Published on: May 11, 2025

This Definite Integral Calculator helps you evaluate definite integrals and shows each step clearly. It works by finding an antiderivative, applying the given upper and lower bounds, and subtracting the results correctly using the Fundamental Theorem of Calculus. This makes it useful for checking answers, understanding how definite integration works, and practising calculus step by step.

Step-by-step method

  1. Set up the definite integral.
  2. Pick the rule.
  3. Plug the problem into the rule.
  4. Solve each integral piece.
  5. Evaluate using F(b) − F(a).
  6. Combine results.

Formula:

[F(x)]ba = F(b) − F(a)
∫ab(u ± v)dx = ∫abu dx ± ∫abv dx
∫abk·u dx = k·∫abu dx
∫ xn dx =
xn+1
n+1
∫
1
x
dx = ln|x|
∫ ex dx = ex
∫ sin(x) dx = −cos(x)
∫ cos(x) dx = sin(x)
∫ x dx =
x2
2
∫ c dx = c·x
Rule: computed symbolically

Example 1: x^2 + 1, x, 0, 2

Step 1 - Set up the definite integral.

In this problem: Lower bound a = 0, upper bound b = 2.

∫20x2 + 1 dx

Step 2 - Pick the rule.

In this problem: Use the Sum/Difference rule.

∫ab(u ± v)dx = ∫abu dx ± ∫abv dx

Step 3 - Plug into the rule.

In this problem: Split the definite integral across the terms.

∫20x2 + 1 dx = ∫20x2 dx + ∫201 dx

Step 4A - Set up this piece.

In this problem: Work one piece at a time.

∫20x2 dx

Step 4B - Pick the rule.

In this problem: Use Power rule.

∫ xn dx =
xn+1
n+1

Step 4C - Apply the rule.

In this problem: Find an antiderivative F(x).

∫ x2 dx =
1
3
x3

Step 4D - Evaluate.

In this problem: Use F(b) − F(a).

∫20x2 dx = [
1
3
x3]20 =
8
3
 − 0 =
8
3

Step 4E - Set up this piece.

In this problem: Work one piece at a time.

∫201 dx

Step 4F - Pick the rule.

In this problem: Use Constant rule.

∫ c dx = c·x

Step 4G - Apply the rule.

In this problem: Find an antiderivative F(x).

∫ 1 dx = x

Step 4H - Evaluate.

In this problem: Use F(b) − F(a).

∫201 dx = [x]20 = 2 − 0 = 2

Step 5 - Combine results.

In this problem: Add the evaluated pieces together.

8
3
 + 2 =
14
3

Final answer: 14/3

Example 2: sin(x), x, 0, pi

Step 1 - Set up the definite integral.

In this problem: Lower bound a = 0, upper bound b = pi.

∫π0sin(x) dx

Step 2 - Pick the rule.

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

∫ sin(x) dx = −cos(x)

Step 3 - Plug into the rule.

In this problem: Use the identity for this integrand.

∫π0sin(x) dx

Step 4A - Set up this piece.

In this problem: Work one piece at a time.

∫π0sin(x) dx

Step 4B - Pick the rule.

In this problem: Use Trig rule (sin).

∫ sin(x) dx = −cos(x)

Step 4C - Apply the rule.

In this problem: Find an antiderivative F(x).

∫ sin(x) dx = −cos(x)

Step 4D - Evaluate.

In this problem: Use F(b) − F(a).

∫π0sin(x) dx = [−cos(x)]π0 = 1 − -1 = 2

Step 5 - Combine results.

In this problem: Add the evaluated pieces together.

2

Final answer: 2