Evaluate the following integrals : ∫ √(cosec x - 1) dx ​

Question

Evaluate the following integrals :

\(\int\sqrt{cosec\,x-1}\)dx

Answer 1

\(\int\sqrt{cosec\,x-1}\)dx

Since, 

cosec x = 1/ sinx

Multiply with (1 + sinx) both numerator and denominator

Since, 

(a + b) × (a - b) = a² - b²,

Let sinx = t 

dt = cosx dx 

Therefore,

Multiply and divide by 2 and add and subtract (1/2)² in denominator,

Let t + 1/2 = u 

dt = du