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 radical expressions:

Substitution 1. If the integrand contains a² − x², let

and use the identity

Substitution 2. If the integrand contains a² + x², let

and use the identity

${\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta .}$

Substitution 3. If the integrand contains x² − a², let

and use the identity

In the integral

we may use

Note that the above step requires that a > 0 and cos(θ) > 0; we can choose the a to be the positive square root of a²; and we impose the restriction on θ to be −π/2 < θ < π/2 by using the arcsin function.

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

Some care is needed when picking the bounds. The integration above requires that −π/2 < θ < π/2, so θ going from 0 to π/6 is the only choice. If we had missed this restriction, we might have picked θ to go from π to 5π/6, which would give us the negative of the result.

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