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

Question

Evaluate:  

\(\int\limits_0^{\pi/2}(\sqrt{tan\,x}+\sqrt{cot\,x})dx\)

∫ (√tan x+√cot x)dx, x ∈[0,π/2]

Answer 1

Let \(I=\int\limits_0^{\pi/2}(\sqrt{tan\,x}+\sqrt{cot\,x})dx\)

=\(\int\limits_0^{\pi/2}\cfrac{sin\,x+cos\,x}{\sqrt{sinxcosx}}dx\)

Let sin x- cos x=t 

⇒ (cos x + sin x)dx=dt.