Definite Integral Calculator

Published on: July 18, 2026

Final Answer: Free • Full Steps: Plus

This Definite Integral Calculator evaluates the integral of a function over an interval [a, b]. It finds the antiderivative using the matching integration rule, then applies the Fundamental Theorem of Calculus: F(b) − F(a). Because the constant of integration cancels, definite integrals return a single number.

Step-by-step method

Set up the definite integral with its limits.
Find the antiderivative F(x) using the matching integration rule.
Apply the Fundamental Theorem of Calculus: evaluate F(b) - F(a).
Simplify to the final value.

Formula:

\(\int_{a}^{b} f\left(x\right) \, dx = F\left(b\right) - F\left(a\right)\)

Example 1:

\(\int_{0}^{3} x^{2} \, dx\)

Step 1 - Set up the definite integral with its limits.

In this problem: We are evaluating \(x^{2}\) from \(x = 0\) to \(x = 3\).

\(\int_{0}^{3} x^{2} \, dx\)

Step 2 - Find the antiderivative F(x) using the matching integration rule.

In this problem: Using the power rule for integration, the antiderivative is \(\frac{x^{3}}{3}\).

\(F\left(x\right) = \frac{x^{3}}{3}\)

Problem

\(\int x^{2}\, dx\)

Approach

Step-by-step method

Identify the exponent n (and any constant multiple).
Write the power rule for integration.
Raise the exponent by one and divide by the new exponent.
Simplify and add the constant of integration C.

Step 1

Step 1 - Identify the exponent n (and any constant multiple).

In this problem: The exponent is \(n = 2\).

\(\int x^{2}\, dx\)

Step 2

Step 2 - Write the power rule for integration.

In this problem: Raise the exponent by one, then divide by the new exponent.

\(\int x^{n}\, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1\)

Step 3

Step 3 - Raise the exponent by one and divide by the new exponent.

In this problem: The new exponent is \(2 + 1 = 3\).

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

Step 4

Step 4 - Simplify and add the constant of integration C.

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

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

Final Answer

\(\frac{x^{3}}{3} + C\)

Step 3 - Apply the Fundamental Theorem of Calculus: evaluate F(b) - F(a).

In this problem: Substitute the upper and lower limits, then subtract.

\(\int_{0}^{3} x^{2} \, dx = \left[ \frac{x^{3}}{3} \right]_{0}^{3} = \left(9\right) - \left(0\right)\)

Step 4 - Simplify to the final value.

In this problem: The value of the definite integral is \(9\).

\(\int_{0}^{3} x^{2} \, dx = 9\)

Final answer:

\(\int_{0}^{3} x^{2} \, dx = 9\)

Example 2:

\(\int_{1}^{4} 2 x + 1 \, dx\)

Step 1 - Set up the definite integral with its limits.

In this problem: We are evaluating \(2 x + 1\) from \(x = 1\) to \(x = 4\).

\(\int_{1}^{4} 2 x + 1 \, dx\)

Step 2 - Find the antiderivative F(x) using the matching integration rule.

In this problem: Using the sum rule for integration, the antiderivative is \(x^{2} + x\).

\(F\left(x\right) = x^{2} + x\)

Problem

\(\int 2 x + 1\, dx\)

Approach

Step-by-step method

Set up the terms in the sum.
Write the sum rule for integration.
Split the integral across the addition signs.
Integrate each separated term.
Combine the results and add the constant of integration C.

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are integrating a sum. The separate terms are \(2 x\), \(1\).

\(\int 2 x + 1\, dx\)

Step 2

Step 2 - Write the sum rule for integration.

In this problem: The integral of a sum is the sum 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 addition signs.

In this problem: Apply the integral to each separated term.

\(\int \left(2 x + 1\right)\, dx = \int 2 x\, dx + \int 1\, dx\)

Step 4

Step 4 - Integrate each separated term.

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

\(\int 2 x\, dx + \int 1\, dx = x^{2} + x + C\)

Step 5

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

In this problem: The combined antiderivative is \(x^{2} + x\).

\(\int \left(2 x + 1\right)\, dx = x^{2} + x + C\)

Final Answer

\(x^{2} + x + C\)

Step 3 - Apply the Fundamental Theorem of Calculus: evaluate F(b) - F(a).

In this problem: Substitute the upper and lower limits, then subtract.

\(\int_{1}^{4} 2 x + 1 \, dx = \left[ x^{2} + x \right]_{1}^{4} = \left(20\right) - \left(2\right)\)

Step 4 - Simplify to the final value.

In this problem: The value of the definite integral is \(18\).

\(\int_{1}^{4} 2 x + 1 \, dx = 18\)

Final answer:

\(\int_{1}^{4} 2 x + 1 \, dx = 18\)