your first answer is correct
While I do some investigation of the complicated induction work on no. 335 ...
No. 352 in my list (there were a load of easy sine formulas):
which is supposed to work out to:
but is going a bit astray.
I tried Parts, with and got so far:
The last part is actually not as bad:
S cos^2(ax)/sin^3(ax) dx right? Rewrite as
S cot^2(ax)*csc(ax) dx
Letting u=csc(ax), we have du=-a*cot^2(ax)dx. Then, cot^2(ax)dx = (-1/a)du Now the integral becomes
S u*(-1/a)du = (-1/a) S u du = (-1/a) u^2/2 = (-1/a) csc^2(ax)/2 = (-1/2a) csc^2(ax) = -1/[2a*sin^2(ax)].
Hmm... the natural logarithm doesn't show up anywhere. Maybe you should have started with S csc^3(ax) dx instead...