Question

Find the integral of the function: $\frac{{\sin }^{3}x+{\cos }^{3}x}{{\sin }^{2}x{\cos }^{2}x}$

Answer 1

$\int \left(\frac{{\sin }^{3}x+{\cos }^{3}x}{{\sin }^{2}x{\cos }^{2}x}\right).dx$

$=\int (\frac{{\sin }^{3}x}{{\sin }^{2}x{\cos }^{2}x}+\frac{{\cos }^{3}x}{{\sin }^{2}x{\cos }^{2}x}).dx$

$=\int (\frac{\sin x}{{\cos }^{2}x}+\frac{\cos x}{{\sin }^{2}x}).dx$

$=\int (\frac{\sin x}{\cos x}.\frac{1}{\cos x}+\frac{\cos x}{\sin x}.\frac{1}{\sin x}).dx$

$=\int \left[\right(\tan x.\sec x.)+(\cot x.\text{cosec x}\left)\right].dx$

$=\int \sec x\tan x.dx+\int \cot x.\text{cosec x}.dx$

$=\sec x+(-\text{cosec x})+C$

$=\sec x-\text{cosec x}+C$

Where $C$ is constant of integration.