Partial Fraction Decomposition for Integrating Rational Functions

[JamesGoh]

Im trying to find the integral of ( sec(t)^2 ) / ( (tan(t)^3) + (tan(t)^2) ). I've managed to get the
integral into the form

1 / (u^3 + u^2) where u = tan(t), however I am having difficulty proceeeding from there.

Could someone take a look at the working out I have attached and let me know what I am not doing right? (the correct answer is written in red pen on 2nd page)

 

[gopher_p]

In order to evaluate $\int\frac{1}{u^2(u+1)}\ du$, you want to use partial fraction decomposition, and that alone. You do not need to do any integration by parts. You have a correct general form $$\frac{1}{u^2(u+1)}=\frac{A}{u^2}+\frac{B}{u+1}+\frac{C}{u}$$ for the PFD, but then it looks like it all goes south after that, and you just gave up on that idea. Stick with that Idea.