Evaluate the following integral: ∫(√(cot x))/(√(tan x)+ √(cot x))dx, x ∈ [0, π/2]

Question

Evaluate the following integral:

\(\int\limits_{0}^{\pi/2}\cfrac{\sqrt{cot\text x}}{\sqrt{tan\text x}+\sqrt{cot\text x}}d\text x \)

Answer 1

Let us assume

I = \(\int\limits_{0}^{\pi/2}\cfrac{\sqrt{cot\text x}}{\sqrt{tan\text x}+\sqrt{cot\text x}}d\text x \)....equation 1

By property, we know that

Thus in equation 2

Adding equation 1 and 2

We know

Thus