Integration by trigonometric substitution is rather complicated. We start by looking at the radical expression in the denominator √(9-25x2) and choose our substitution so that we can get 1-sin2θ = cos2θ. We can choose x = 3/5 sinθ (and dx = 3/5 cosθdθ) so that √(9-25x2) becomes 3cosθ after the substitution.
Without giving you every step or the answer, you should be able to get to a point where you are integrating 5/9 csc2θdθ after the cosines cancel. You then need to integrate and go back to use a right triangle containing the original substitution x = 3/5 sinθ to determine the final answer. Hope this helps!
