Evaluate the following integral: ∫√ (tan x)/√ (tan x) + √(cot x)dx, x ∈ [π/6, π/3]

Question

Evaluate the following integral: 

\(\int\limits_{\pi/6}^{\pi/3}\cfrac{\sqrt{tan\text x}}{\sqrt{tan\text x}+ \sqrt{cot\text x}}d\text x \)

Answer 1

Let us assume,

I = \(\int\limits_{\pi/6}^{\pi/3}\cfrac{\sqrt{tan\text x}}{\sqrt{tan\text x}+ \sqrt{cot\text x}}d\text x \)..... equation 1

By property, we know that

Trigonometric property

I = \(\int\limits_{\pi/6}^{\pi/3}\cfrac{\sqrt{cot\text x}}{\sqrt{cot\text x}+ \sqrt{tan\text x}}d\text x \).....equation 2

Adding equation 1 and equation 2

We know

where b and a are upper and lower limits respectively and f(x) is a function