Question

is there any way to integrate sec^3 without using integration by parts?

Answer 1

\(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\sec^3 x~dx}\)

\(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{\cos x}{(1-\sin^2 x)^2}~dx}\)

u= sin x
du = cos(x) dx

\(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{1}{(1-u^2)^2}~du}\)
http://www.wolframalpha.com/input/?i=partial+fractions+1%2F%28%281-u%5E2%29%5E2%29

then continue on from partial fraction decomposition (that I show in wolfram)