Definite Integrals - 6

1. The value of the integral ∫₀ ^{π /2} [√(cot x)/[ √(cot x) +√(tan x)]]dx is

a. π/4
b. π/2
c. π
d. none of these
(JEE 1983)

Answer: (a)

Solution:

Concept from definite integration to be used:

∫₀^af(x) = ∫₀ ^af(x-a)


I = ∫₀ ^{π /2} [√(cot x)/[ √(cot x) +√(tan x)]]dx ….(i)

= ∫₀ ^{π /2} [√(cot(π /2- x))/[ √(cot (π /2 -x)) +√(tan(π /2- x))]]dx

= ∫₀ ^{π /2} [√(tan x)/[ √(tan x) +√(cot x)]]dx … (ii)

Adding (i) and (ii)

2I = ∫₀ ^{π /2} [[√(cot x) +√(tan x)]/[ √(tan x) +√(cot x)]]dx

2I = ∫₀ ^{π /2}dx = π /2
I = π /4