Trigonometric substitution

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, due to the fact that these trigonometric identities reduce two terms to a single term.

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

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In the integral

one may write

so that the integral becomes

(provided a > 0).