evaluate the integral of 1/cosx(sinx)^5
Under the assumption that the function we have to integrate is…
$\displaystyle f(x)= \frac {1}{\cos x \cdot sin^{5} x}$
… the ‘standard approach’ is the change of variable…
$\displaystyle t=\tan \frac {x}{2}$
… so that...
$\displaystyle \sin x = \frac {2t}{1+t^{2}}, \cos x= \frac {1-t^{2}}{1+t^{2}}, dx= 2 \cdot \frac {dt}{1+t^{2}}$
... and we arrive to an integral that contains rational functions and that can be solved without great difficulty. Other substitutions doesn’t work so well … at least in my opinion…
Regards