Question

$\int \frac{{\sin }^{3}x+{\cos }^{3}x}{{\sin }^{2}x{\cos }^{2}x}dx$ is equal to

• A. $\sec x-cosecx+c$
• B. $\sec x+cosecx+c$
• C. $\sin x-\cos x+c$
• D. $\sin x+\cos x+c$

Answer 1

The correct option is A $\sec x-cosecx+c$

$I=\int \frac{{\sin }^{3}x+{\cos }^{3}x}{{\sin }^{2}x{\cos }^{2}x}dx$

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

$=\int \tan x.\sec xdx+\int \cot xcosecx.dx$

We know that,

$\int \tan x\sec x.dx=\sec x+c$ and $\int \cot xcosecxdx=-\text{cosec}x+c$

So,

$I=\sec x-\text{cosec}x+c$