The value of the integral int_0^π/2 dfrac√(cot x)√(cot x) + √(tan x) dx is

The value of the integral int_0^π/2 dfrac√(cot x)√(cot x) +… — JEE Main 1983 Mathematics PYQ

1983

The value of the integral

\int_0^{\pi/2} \dfrac{\sqrt{\cot x}}{\sqrt{\cot x} + \sqrt{\tan x}}\, dx

A.
$\pi/4$
B.
$\pi/2$
C.
$\pi$
D.
None of these

Show answer and solution

Correct option: A

Solution

Using the property ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx, let I = ∫₀^(π/2) √(cot x)/(√(cot x)+√(tan x)) dx. Replacing x with (π/2 − x) swaps cot x and tan x, giving I = ∫₀^(π/2) √(tan x)/(√(tan x)+√(cot x)) dx. Adding both forms: 2I = ∫₀^(π/2) 1 dx = π/2, so I = π/4.

[1983]

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