When the integrand includes certain radical expressions, a set of trigonmetric substitutions can be used. These substitutions change the equation from one that is a function of 'x' into trigonometric functions of angle Θ. Once expressed in terms of Θ, variable values can be determined using the basic trigonometric identities.
We start by following a 'recipe' of the substitution rule, solve the integration in terms of Θ, then replace the original variables (in terms of 'x') into the result. For example, here we evaluate:
$\int \frac{dx}{{x}^{2}\sqrt{9-{x}^{2}}}$