In mathematics, trigonometric substitution is the replacement of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identities to simplify certain integrals containing radical expressions. Like other methods of integration by substitution, when evaluating a definite integral, it may be simpler to completely deduce the antiderivative before applying the boundaries of integration.
Let and use the identity
In the integral
we may use
Then,
The above step requires that and We can choose to be the principal root of and impose the restriction by using the inverse sine function.
For a definite integral, one must figure out how the bounds of integration change. For example, as goes from to then goes from to so goes from to Then,
Some care is needed when picking the bounds. Because integration above requires that , can only go from to Neglecting this restriction, one might have picked to go from to which would have resulted in the negative of the actual value.
Alternatively, fully evaluate the indefinite integrals before applying the boundary conditions. In that case, the antiderivative gives
as before.
The integral
may be evaluated by letting where so that and by the range of arcsine, so that and
Then,
For a definite integral, the bounds change once the substitution is performed and are determined using the equation with values in the range Alternatively, apply the boundary terms directly to the formula for the antiderivative.
For example, the definite integral
may be evaluated by substituting with the bounds determined using
Because and
On the other hand, direct application of the boundary terms to the previously obtained formula for the antiderivative yields
as before.
Let and use the identity
In the integral
we may write
so that the integral becomes
provided
For a definite integral, the bounds change once the substitution is performed and are determined using the equation with values in the range Alternatively, apply the boundary terms directly to the formula for the antiderivative.
