Question

Find the integrals of the functions $\frac{{\sin }^{3}x+{\cos }^{3}x}{{\sin }^{2}x{\cos }^{2}x}$

Answer 1

$\frac{{\sin }^{3}x+{\cos }^{3}x}{{\sin }^{2}x{\cos }^{2}x}=\frac{{\sin }^{3}x}{{\sin }^2{\cos }^{2}x}+\frac{{\cos }^{3}x}{{\sin }^{2}x{\cos }^{2}x}$
$=\frac{\sin x}{{\cos }^{2}x}+\frac{\cos x}{{\sin }^{2}x}$
$=\tan x\sec x+\cot xco\sec x$
$\therefore \int \frac{{\sin }^{3}x+{\cos }^{3}x}{{\sin }^{2}x{\cos }^{2}x}dx=\int (\tan x\sec x+\cot xco\sec x)dx$
$=\sec x-co\sec x+C$