Understanding Integrals and Antiderivatives: A Complete Guide

Integration is the reverse of differentiation. Where derivatives tell us how a function changes, integrals tell us how those changes accumulate. Whether you are computing the area under a curve, the total distance travelled, or the net change in a quantity, integration is the tool you need.

This guide covers the two types of integrals, the fundamental rules, and several worked examples to build your confidence.

Indefinite Integrals (Antiderivatives)

An indefinite integral finds the general antiderivative of a function. If $F'(x) = f(x)$ , then:

\int f(x)\,dx = F(x) + C

The constant $C$ is called the constant of integration. It accounts for the fact that infinitely many functions share the same derivative.

Definite Integrals

A definite integral computes the net signed area between a function and the x-axis over an interval $[a, b]$ . The Fundamental Theorem of Calculus connects the two types:

\int_a^b f(x)\,dx = F(b) - F(a)

Here $F$ is any antiderivative of $f$ . You evaluate $F$ at the upper and lower limits and subtract.

Basic Integration Rules

The most important rules mirror the derivative rules in reverse:

Power rule: $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$ )
Constant multiple: $\int k \cdot f(x)\,dx = k \int f(x)\,dx$
Sum/difference: $\int [f(x) \pm g(x)]\,dx = \int f(x)\,dx \pm \int g(x)\,dx$
Exponential: $\int e^x\,dx = e^x + C$
Reciprocal: $\int \frac{1}{x}\,dx = \ln|x| + C$

Worked Examples

Example 1: Power Rule

Find $\int 4x^3\,dx$ .

Apply the power rule and constant multiple rule:

\int 4x^3\,dx = 4 \cdot \frac{x^{3+1}}{3+1} + C = 4 \cdot \frac{x^4}{4} + C = x^4 + C

Example 2: Polynomial Integration

Find $\int (6x^2 - 3x + 7)\,dx$ .

Integrate each term separately:

\int (6x^2 - 3x + 7)\,dx = 6 \cdot \frac{x^3}{3} - 3 \cdot \frac{x^2}{2} + 7x + C = 2x^3 - \frac{3x^2}{2} + 7x + C

Example 3: Definite Integral

Evaluate $\int_1^3 (2x + 1)\,dx$ .

First find the antiderivative:

F(x) = x^2 + x

Then evaluate at the bounds:

F(3) - F(1) = (9 + 3) - (1 + 1) = 12 - 2 = 10

Example 4: Exponential and Reciprocal

Find $\int \left(e^x + \frac{2}{x}\right)\,dx$ .

\int \left(e^x + \frac{2}{x}\right)\,dx = e^x + 2\ln|x| + C

Integration by Substitution

When the integrand contains a composite function, substitution (the reverse of the chain rule) is often the right approach. Choose $u = g(x)$ , compute $du = g'(x)\,dx$ , and rewrite the integral in terms of $u$ .

Example 5: U-Substitution

Find $\int 2x \cdot e^{x^2}\,dx$ .

Let $u = x^2$ , so $du = 2x\,dx$ .

\int 2x \cdot e^{x^2}\,dx = \int e^u\,du = e^u + C = e^{x^2} + C

Common Mistakes to Avoid

Forgetting the constant of integration. Every indefinite integral must include $+ C$ .
Misapplying the power rule when n = -1. The integral of $x^{-1}$ is $\ln|x| + C$ , not $\frac{x^0}{0}$ .
Swapping limits. Reversing the limits of a definite integral changes the sign of the result.

Try It Yourself

Practise integration problems with our free Integral Calculator. It shows the full solution with each step, so you can follow along and master the technique. For evaluating integrals with specific bounds, use our Definite Integral Calculator.