Definite Integral Notes

Published on: September 29, 2026

This Definite Integral Calculator evaluates an integral between two limits, and shows the working line by line. The antiderivative is found first with the matching integration rule, and the Fundamental Theorem then turns it into a single number by subtracting its value at the lower limit from its value at the upper one. No constant of integration is needed, because it cancels in that subtraction. Following the same pattern every time keeps the two limits the right way round.

Step-by-step method

Set up the problem.
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:

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

See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

Set up the problem.
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.

Step 1

Step 1 - Set up the problem.

In this problem:

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

Step 2

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

In this problem:

\(F\left(x\right) = \frac{x^{4}}{4}\)

Problem

\(\int x^{3}\, dx\)

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 \(n = 3\).

\(\int x^{3}\, dx\)

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 \(3 + 1 = 4\).

\(\int x^{3}\, dx = \frac{x^{4}}{4} + C\)

Step 4

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

In this problem: The simplified antiderivative is \(\frac{x^{4}}{4}\).

\(\int x^{3}\, dx = \frac{x^{4}}{4} + C\)

Final Answer

\(\frac{x^{4}}{4} + C\)

Step 3

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

In this problem:

\(\int_{0}^{2} x^{3} \, dx = \left[ \frac{x^{4}}{4} \right]_{0}^{2} = \left(4\right) - \left(0\right)\)

Step 4

Step 4 - Simplify to the final value.

In this problem:

\(\int_{0}^{2} x^{3} \, dx = 4\)

Final Answer

\(\int_{0}^{2} x^{3} \, dx = 4\)

See Example 2 Hide Example 2

Problem

\(\int_{1}^{5} 4 x + 3 \, dx\)

Approach

Step-by-step method

Set up the problem.
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.

Step 1

Step 1 - Set up the problem.

In this problem:

\(\int_{1}^{5} 4 x + 3 \, dx\)

Step 2

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

In this problem:

\(F\left(x\right) = 2 x^{2} + 3 x\)

Problem

\(\int 4 x + 3\, dx\)

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 \(4 x\), \(3\).

\(\int 4 x + 3\, dx\)

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.

\(\int \left(4 x + 3\right)\, dx = \int 4 x\, dx + \int 3\, dx\)

Step 4

Step 4 - Integrate each separated term.

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

\(\int 4 x\, dx + \int 3\, dx = 2 x^{2} + 3 x + C\)

Step 5

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

In this problem: The combined antiderivative is \(2 x^{2} + 3 x\).

\(\int \left(4 x + 3\right)\, dx = 2 x^{2} + 3 x + C\)

Final Answer

\(2 x^{2} + 3 x + C\)

Step 3

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

In this problem:

\(\int_{1}^{5} 4 x + 3 \, dx = \left[ 2 x^{2} + 3 x \right]_{1}^{5} = \left(65\right) - \left(5\right)\)

Step 4

Step 4 - Simplify to the final value.

In this problem:

\(\int_{1}^{5} 4 x + 3 \, dx = 60\)

Final Answer

\(\int_{1}^{5} 4 x + 3 \, dx = 60\)

See Example 3 Hide Example 3

Problem

\(\int_{0}^{3} x^{2} + 1 \, dx\)

Approach

Step-by-step method

Set up the problem.
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.

Step 1

Step 1 - Set up the problem.

In this problem:

\(\int_{0}^{3} x^{2} + 1 \, dx\)

Step 2

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

In this problem:

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

Problem

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

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 \(x^{2}\), \(1\).

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

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.

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

Step 4

Step 4 - Integrate each separated term.

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

\(\int x^{2}\, dx + \int 1\, dx = \frac{x^{3}}{3} + x + C\)

Step 5

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

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

\(\int \left(x^{2} + 1\right)\, dx = \frac{x^{3}}{3} + x + C\)

Final Answer

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

Step 3

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

In this problem:

\(\int_{0}^{3} x^{2} + 1 \, dx = \left[ \frac{x^{3}}{3} + x \right]_{0}^{3} = \left(12\right) - \left(0\right)\)

Step 4

Step 4 - Simplify to the final value.

In this problem:

\(\int_{0}^{3} x^{2} + 1 \, dx = 12\)

Final Answer

\(\int_{0}^{3} x^{2} + 1 \, dx = 12\)

See Example 4 Hide Example 4

Problem

\(\int_{1}^{4} 6 x - 2 \, dx\)

Approach

Step-by-step method

Set up the problem.
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.

Step 1

Step 1 - Set up the problem.

In this problem:

\(\int_{1}^{4} 6 x - 2 \, dx\)

Step 2

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

In this problem:

\(F\left(x\right) = 3 x^{2} - 2 x\)

Problem

\(\int 6 x - 2\, dx\)

Approach

Step-by-step method

Set up the terms in the difference.
Write the difference rule for integration.
Split the integral across the subtraction 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 difference.

In this problem: We are integrating a difference. The separated terms are \(6 x\), \(2\).

\(\int 6 x - 2\, dx\)

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.

\(\int \left(6 x - 2\right)\, dx = \int 6 x\, dx - \int 2\, dx\)

Step 4

Step 4 - Integrate each separated term.

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

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

Step 5

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

In this problem: The combined antiderivative is \(3 x^{2} - 2 x\).

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

Final Answer

\(3 x^{2} - 2 x + C\)

Step 3

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

In this problem:

\(\int_{1}^{4} 6 x - 2 \, dx = \left[ 3 x^{2} - 2 x \right]_{1}^{4} = \left(40\right) - \left(1\right)\)

Step 4

Step 4 - Simplify to the final value.

In this problem:

\(\int_{1}^{4} 6 x - 2 \, dx = 39\)

Final Answer

\(\int_{1}^{4} 6 x - 2 \, dx = 39\)