35th  Top calculus topics 
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:
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In the integral
we may use
so that the integral becomes
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 a^{2}; 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 result in the negative of the result.
In the integral
we may write
so that the integral becomes
(provided a ≠ 0).
Integrals like
should be done by partial fractions rather than trigonometric substitutions. However, the integral
can be done by substitution:
We can then solve this using the formula for the integral of secant cubed.
Substitution can be used to remove trigonometric functions. For instance,
(but be careful with the signs)
