Trigonometric substitution

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Topics in calculus

Function | Limits of functions | Continuity | Calculus with polynomials

Product rule | Quotient rule | Chain rule | Implicit differentiation | Taylor's theorem

In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities

to simplify certain integrals containing the radical expressions

respectively.

In the expression a² + x², the substitution of a tan(θ) for x makes it possible to use the identity tan²θ + 1 = sec²θ.

Similarly, in x² − a², the substitution of sec(θ) for x makes it possible to use the identity sec² − 1 = tan².

Examples

In the integral

one may use

so that the integral becomes

(provided a > 0; if a < 0 then √a² would be |a|, which would differ from a).

For a definite integral, one must figure out how the bounds of integration change. For example, as x goes from 0 to a/2, then sin(θ) goes from 0 to 1/2, so θ goes from 0 to π/6. Then we have

In the integral

one may write

so that the integral becomes

(provided a > 0).