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Master Definite Integrals

Complete guide to definite integrals with free interactive practice. Learn the Fundamental Theorem of Calculus, evaluate integrals with bounds, and find area under curves.

What are Definite Integrals?

Understanding integrals with bounds and the Fundamental Theorem of Calculus

A definite integral ∫ₐᵇ f(x) dx represents the signed area under the curve y = f(x) between x = a and x = b. Unlike indefinite integrals, definite integrals produce a number (not a function plus C). The Fundamental Theorem of Calculus states that if F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a). This powerful result connects differentiation and integration and is the main tool for evaluating definite integrals.

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Fundamental Theorem

Learn how ∫ₐᵇ f(x) dx = F(b) − F(a) lets you evaluate any definite integral once you know an antiderivative.

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Area Under Curves

Definite integrals compute the signed area between a curve and the x-axis. Positive where f(x) > 0, negative where f(x) < 0.

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Numeric Answers

Each definite integral evaluates to a single number. Practice evaluating integrals with polynomials, trig, and exponentials.

Definite Integral Rules

Key formulas and properties for evaluating definite integrals

Fundamental Theorem of Calculus (Part 2)

If F is any antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a). First find F, then substitute the bounds and subtract.

\int ₐᵇ f(x) \, dx = F(b) − F(a)
\int ₀¹ x^{2} \, dx = \frac{[x^{3}}{3]₀¹} = \frac{1}{3} − 0 = \frac{1}{3}
\int ₀^π sin(x) \, dx = [-cos(x)]₀^π = 2

Power Rule (with bounds)

For ∫ₐᵇ x^n dx with n ≠ −1: evaluate [x^(n+1)/(n+1)] at b and a, then subtract.

∫ₐᵇ x^n dx = \frac{[x^{n+1}}{n + 1}]ₐᵇ
\int ₀¹ x \, dx = \frac{[x^{2}}{2]₀¹} = \frac{1}{2}
\int ₁² x^{2} \, dx = \frac{[x^{3}}{3]₁²} = \frac{8}{3} − \frac{1}{3} = \frac{7}{3}

Constant and Sum Rules

Constants factor out; you can split integrals over sums. These apply to definite integrals just like indefinite ones.

\int ₐᵇ k f(x) \, dx = k·\int ₐᵇ f(x) \, dx
\int ₐᵇ (f(x) + g(x)) \, dx = \int ₐᵇ f(x) \, dx + \int ₐᵇ g(x) \, dx

Trigonometric Definite Integrals

Common trig definite integrals with standard bounds:

\int ₀^π sin(x) \, dx = 2
\int ₀^(\frac{π}{2}) cos(x) \, dx = 1
\int ₀^π cos(x) \, dx = 0

Exponential Definite Integrals

∫ₐᵇ e^x dx = e^b − e^a. For e^(kx), divide by k when evaluating.

\int ₀¹ e^x \, dx = e − 1 ≈ 1.718
\int ₀¹ e^(2x) \, dx = \frac{e^{2} − 1}{2}

Definite Integral Examples

Worked examples with numeric answers

Basic Examples

\int ₀¹ x \, dx\frac{1}{2}
\int ₀¹ x^{2} \, dx\frac{1}{3}
\int ₀² 3 \, dx6
\int ₀^π sin(x) \, dx2
\int ₀^(\frac{π}{2}) cos(x) \, dx1

Why Definite Integrals Matter

Definite integrals are used everywhere: computing areas, displacement from velocity, total change from a rate, work done by a force, and many applications in physics, economics, and engineering. Mastering them is essential for calculus and beyond.

Applications

Definite integrals compute area under curves, total distance traveled, accumulated change (e.g., total cost from marginal cost), and appear in probability, signal processing, and differential equations.

Start Practicing Definite Integrals

Practice evaluating definite integrals with our infinite generator and instant feedback

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Definite Integral Practice

Practice definite integrals with bounds. Evaluate ∫ₐᵇ f(x) dx and get instant feedback.

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Indefinite Integrals First

Need a refresher on antiderivatives? Practice indefinite integrals before tackling definite integrals.

Practice Indefinite Integrals