Question

Integration of sec^2x.cosec^2x

Answer 1

Answer:

$\displaystyle \longrightarrow \tan x - \cot x + C$

Step-by-step explanation:

We have to integrate as given :

$\displaystyle \int\limits {\sec^2 x.\csc^2 x} \, dx$

We know :

$\displaystyle \csc^2 x=1+\cot^2 x$

Putting values there we get :

$\displaystyle \int\limits {\sec^2 x.(1+\cot^2 x)} \, dx$

$\displaystyle \int\limits {\sec^2 x+\sec^2 x.\cot^2 x} \, dx$

$\displaystyle \int\limits {\sec^2 x+\frac{1}{\cos^2 x} .\frac{\cos^2 x}{\sin^2 x} } \, dx$

Here cos² x get cancel out and we know reciprocal of sin² x is csc² x

$\displaystyle \int\limits {\sec^2 x+\csc^2 x} \, dx$

$\displaystyle \int\limits {\sec^2 x} \, dx+\int\limits {\csc^2 x} \, dx$

$\displaystyle \longrightarrow \tan x + ( - \cot x)$

$\displaystyle \longrightarrow \tan x - \cot x + C$

Hence we get required answer.