Question

$\int \frac{si{n}^{3}x+co{s}^{3}x}{si{n}^{2}x.co{s}^{2}x}dx$

Answer 1

$\int \frac{1}{\sin x{\cos }^{2}x}dx=\int \frac{{\sin }^{2}x+{\cos }^{2}x}{\sin x{\cos }^{2}x}dx$

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

$=\int \frac{\sin x}{{\cos }^{2}x}dx+\int \frac{1}{\sin x}dx$

$=\int \sec x\cdot \tan xdx+\int cosecxdx$

$=\sec x+ln|cosecx-\cot x|+c$.