Section 17

Guide to choosing a method

Use the signal in the integrand to decide which technique to try first.

Signal → method

Signal
Composition and the inner derivative appears
Method
Substitution
Signal
Product with logarithm, inverse trigonometric, or algebraic power
Method
Integration by parts
Signal
Powers of sine and cosine
Method
Identities and parity rules
Signal
Powers of tangent and secant
Method
Reserve $sec^{2} x$ or $sec x tan x$
Signal
Quotient of polynomials
Method
Division and partial fractions
Signal
$a^{2} - x^{2}$ , $a^{2} + x^{2}$ , $x^{2} - a^{2}$
Method
Trigonometric substitution
Signal
Non-standard quadratic inside a radical or denominator
Method
Completing the square
Signal
Infinite limits or singularities
Method
Improper integral using limits
Signal
Difficult antiderivative or tabulated data
Method
Numerical integration
Signal
Integrand with a recurring parameter
Method
Reduction formula

Verify the domain of the integrand.
Identify if the integral is proper or improper.
Maintain consistency of variable and limits during a substitution.
Add $C$ only in indefinite integrals.
In areas and volumes, use nonnegative distances and areas.
In series, first check that $a_{n} \to 0$ , but do not confuse this necessary condition with a sufficient criterion.
Differentiate the obtained antiderivative whenever possible.