Question

Integrate : ∫(1/(1 - sin x)) dx

Answer 1

\(\int \frac 1{1-\sin x}dx\)

\(= \int \frac 1{1- \sin x} \times \left(\frac{1+\sin x}{1+ \sin x}\right)dx\)

\(= \int \frac {1 + \sin x}{1 - \sin^2 x}dx\)

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

\(= \int \left(\frac 1{\cos^2x}+\frac{\sin x}{\cos^2x}\right) dx\)

\(= \int \sec^2 x dx + \int \sec x\tan x dx\)

\(= \tan x + \sec x + c\)

Answer 2

Let

(Multiplying numerator & denominator by 1 + sin x)

∫(sec² x + sec x.tan x)dx

= ∫sec² x dx + ∫sec x.tan x dx

= tan x + sec x + k