Your work is good up to here.

Now take note that appears in your answer. Apply integration by parts again to get some constant multiple of to reappear. At this stage, treat the integral as a variable, and solve the equation for the integral (and don't forget the +C at the end!).

Does this make sense? Can you continue where you left off?

Note that2. Evaluate I = integral(486cot(6x)/sin^2(6x)+18sin(6x)+81) dx

No exactly sure how to go about this problem. I can factor the bottom into (sin(6x)+9)^2. I am using partial fraction decomposition, but the numerator seems to confuse me.

Help is much appreciated!

Thanks to all who contribute.

Now apply the substitution and the reduced integral can be solved using partial fractions.

Does this make sense? Can you try to take it from here?