Integration by substitution

Integration by substitution changes the variable of integration to simplify an integral or match it to a known antiderivative. It reverses the chain rule by replacing a composite expression and its differential together.

Chain rule: A differentiation rule for composite functions: the derivative of a composition combines the outer derivative with the inner derivative. Substitution reverses this rule by absorbing the inner derivative into the differential.
Indefinite integral: A family of antiderivatives of a function, conventionally written with an arbitrary constant of integration. The substitution rule for indefinite integrals returns an antiderivative in the original variable.
Trigonometric substitution: An integration technique that replaces an algebraic variable with a trigonometric expression to simplify radicals. It is a specialized substitution for integrands involving characteristic quadratic radicals.
Integration by parts: An integration rule that rewrites the integral of a product using derivatives of one factor and an antiderivative of another. It targets products suited to the product rule, rather than a composite pattern suited to substitution.
Change-of-variables theorem: A theorem relating integrals under a transformation, with a Jacobian factor accounting for changes in scale. The one-variable substitution rule is its simplest integral-calculus instance.
Change of variables: A transformation that replaces one variable with another while adjusting the quantities expressed in that variable. Substitution applies this transformation to the integrand and differential together.
Integration constant: An arbitrary additive constant included in the general family of antiderivatives. An indefinite integral still requires a constant after substitution and back-substitution.
Gaussian integral: The integral of the Gaussian function over the real line, equal to the square root of pi. A change of variables rescales the Gaussian integral to evaluate related integrals.
Partial fraction decomposition: A method for expressing a rational function as a sum of simpler rational fractions. It simplifies rational integrands by algebraic decomposition instead of changing variables.
Jacobian: The determinant of a transformation’s derivative matrix, measuring local volume scaling. In several variables, the Jacobian generalizes the differential factor used in substitution.

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