Evaluate the following integral: ∫ (1)/(1+cot x)dx, x ∈ [0, π/2]

Question

Evaluate the following integral:

\(\int\limits_{0}^{\pi/2}\cfrac{1}{1+cot\text x}d\text x \)

Answer 1

Let us assume

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

We know that tan x = \(\cfrac{cos\text x}{sin\text x}\)

cot x = \(\cfrac{1}{tan\text x}\)

Substituting the value in equation 1 we have, 

By property, we know that

Thus in equation 2

Adding equation 2 and 3

Thus