I'm having trouble with #2. i've got it down to the part where it would be the integral of 5cos^3(pheta)/sin(pheta). i'm not sure where to go from there. u-sub won't work and neither will taking out a tan(pheta). .

Setting , you have . Then the integral becomes






Now, in general. But since we want our substitution to be invertible, we are tacitly assuming that we're working over a restricted domain. In particular, this means , which implies that , or equivalently that . Over this domain, , so .

Long story short, this allows us to go from



to




Computing the remaining integral isn't difficult. Expand the numerator with the Pythagorean identity to get



Then integrate term-by-term to get




Now undo the substitution to get the antiderivative back in terms of .



and using basic trigonometric properties (e.g. Pythagorean theorem) this reduces to