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Mathematics Question on Some Properties of Definite Integrals

The value of the integral ∫0π/2cot⁡ xcot⁡ x+tan⁡ x dx is\int^{ \pi / 2 }_ 0 \frac {\sqrt { \cot \, x }}{ \sqrt { \cot \, x } + \sqrt { \tan \, x }} \, dx \, is

A

π4\frac{\pi}{ 4 }

B

π2\frac{\pi}{ 2 }

C

π\pi

D

None of these

Answer

π4\frac{\pi}{ 4 }

Explanation

Solution

Let  I=∫0π/2cot xcot x+tan x dx is..................(1) Let \, \, I = \int^{ \pi / 2 }_ 0 \frac {\sqrt { cot \, x }}{ \sqrt { cot \, x } + \sqrt { tan \, x }} \, dx \, is ..................(1)
⇒  I=∫0π/2tan xcot x+tan x dx is..................(2)\Rightarrow \, \, I = \int^{ \pi / 2 }_ 0 \frac {\sqrt { tan \, x }}{ \sqrt { cot \, x } + \sqrt { tan \, x }} \, dx \, is ..................(2)
On adding Eqs. (i) and (ii), we get
2I=∫0π/2 1 dx2I = \int^{ \pi / 2 }_ 0 \, 1 \, dx
∴I=π4\therefore I = \frac{\pi}{4 }