Question

Find the integrals of the function:

`1/(sin xcos^3 x)`

Answer 1

Let `I = int 1/(sin x cos^3 x)  dx`

`= int (sin^2 x + cos^2 x)/(sin x cos^3 x)  dx`

`= int ((sin^2 x)/(sin x cos^3 x) + (cos^2 x)/(sin x cos^3 x))  dx`

`= int ((sin x)/(cos 3 x) + (cos x)/(sin x cos^2 x))  dx`

`= int (tan x sec^2 x + (sec^2 x)/(tan x))  dx`

`= int (tan x + 1/(tan x)) sec^2 x  dx`

Put tan x = t

⇒ sec² x dx = dt

∴ `I = int (t + 1/t) dt `

`= t^2/2 + log |t| + C`

`log |tan x| + 1/2 tan^2 x + C`