Integration Guide

Indefinite Integrals Guide

A compact roadmap of the most common techniques for computing $\int f(x)\,dx$ with examples and links to deeper articles.

Definition 0.1: Antiderivative and Indefinite Integral

A function $F$ is an antiderivative of $f$ on an interval if $F'(x)=f(x)$ for all $x$ in the interval.

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

The constant $C$ captures the family of antiderivatives.

Technique Index

1) Substitution: turn a composite into a simple integral.

2) Integration by Parts: product integrals via

\int u\,dv = uv - \int v\,du

3) Rational Functions: partial fractions.

4) Trigonometric Integrals: identities and substitution.

What to Practice

Recognize patterns (chain rule reverse, product rule reverse).

Choose substitutions that simplify the differential.

Use algebraic simplification early (factor/expand if helpful).

Differentiate your answer to verify.

Example 1.1: Basic Pattern Recognition

Compute: $\int (3x^2-4x+1)\,dx$

= x^3 - 2x^2 + x + C

Example 2.1: Substitution (u-substitution)

Compute: $\int 2x\,(x^2+1)^5\,dx$

u=x^2+1,\;du=2x\,dx

\int 2x\,(x^2+1)^5\,dx = \int u^5\,du = \frac{u^6}{6}+C = \frac{(x^2+1)^6}{6}+C

Example 3.1: Integration by Parts

Compute: $\int x e^x\,dx$

u=x,\;dv=e^x\,dx\;\Rightarrow\;du=dx,\;v=e^x

\int x e^x\,dx = x e^x - \int e^x\,dx = x e^x - e^x + C

Remark 0.2: Next Steps

For rational functions, see:Partial Fractions Method.

For trig techniques, see:Trigonometric Integrals, and for the advanced classification approach:Trigonometric Rational Integrals.