Integration techniques

This chapter is a toolkit for common classes of integrals with elementary antiderivatives, not a classification of every elementary integral. Each lesson identifies the structure in the integrand that makes its method work, so choose a method from that structure rather than from the presence of a particular symbol. The chapter assumes substitution, integration by parts, polynomial factorisation, completing the square and the standard trigonometric identities.

Lessons

1. Trigonometric integrals: powers and products of sine, cosine, tangent and secant, using the Pythagorean, double-angle and product-to-sum identities.
2. Reduction formulae: recurrences that organise repeated integration by parts when the same power appears with a lower index.
3. Trigonometric and hyperbolic substitutions: clearing a quadratic surd by matching it to a Pythagorean identity.
4. Rational functions: division and partial fractions, which lead to logarithms and inverse tangents.
5. Rationalising substitutions: changes of variable that turn other surds and exponentials into rational functions.
6. Seen questions: past exam questions combining product-to-sum identities, reduction formulae, substitutions and partial fractions.

Where it leads

The resulting antiderivatives are building blocks in first-order methods for ordinary differential equations and in the iterated integrals of Fubini's theorem.

Vocabulary checkpoints

• Antiderivative, indefinite and definite integral: an antiderivative (or primitive) of $f$ is a function $F$ with $F'=f$; the indefinite integral $\int f(x)\,dx=F(x)+C$ is the family of them on an interval; the definite integral $\int_a^b f(x)\,dx$ is a number, equal to $F(b)-F(a)$ when $f$ is continuous on $[a,b]$ and $F'=f$ there.
• Interval and constant: every indefinite formula holds on a connected interval where the integrand is defined, so the integration constant may be chosen independently on different sides of a pole (rational functions).
• Proper and improper: an improper rational function needs division before partial fractions apply (rational functions).
• Recurrence and antiderivative: a reduction formula relates integrals in a family rather than evaluating one directly (reduction formulae).
• Substitution and branch: a substitution needs a real domain and a branch on which it is one-to-one and the branch fixes the sign of each square root (trigonometric and hyperbolic substitutions).