Need help with trigonometric substitution for integrals

Question

What is the integral of x^3/sqrt(9+x^2) using trigonometric substitution x=3tan(theta). How to go about this?

Answer 1

First make the substitution x=3 tan θ which will result in 27 tan³θ 3 sec²θ dθ/3 sec θ.

Reduce this to sin³θ dθ/cos⁴θ.

Then further substitute u=cos θ which will result in a simple power function integral.

Don't forget to back track through your substitutions to get the answer in terms of x.