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UPSC NDA Math Model Paper 3

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Question : 12 of 120

Marks: +1, -0

The value of the integral

\int\limits_{0}^{\pi/2} \frac{\sqrt{\tan x}}{1+\sqrt{\tan x}} dx

is

$\frac{\pi}{2}$
$\frac{\pi}{4}$
$\frac{\pi}{6}$
none of these

Solution:

Here, we have to find the value of the integral

\int\limits_{0}^{\pi/2} \frac{\sqrt{\tan x}}{1+\sqrt{\tan x}}

Let

I=\int\limits_{0}^{\pi/2} \frac{\sqrt{\tan x}}{1+\sqrt{\tan x}} dx

......(1)

As we know that,

\int\limits_{a}^{b} f(x) d x=\int\limits_{a}^{b} f(a+b-x) d x

I=\int\limits_{0}^{\pi/2} \frac{\sqrt{\cot x}}{1+\sqrt{\cot x}} dx

......(2)

Adding equation (1) and (2), we get

\Rightarrow 2I=\int\limits_{0}^{\pi/2} \left[ \frac{\sqrt{\tan x}}{1+\sqrt{\tan x}}+\frac{\sqrt{\cot x}}{1+\sqrt{\cot x}} \right] d x

\Rightarrow 2I=\int\limits_{0}^{\pi/2} d x=\frac{\pi}{2}

\Rightarrow I = \frac{\pi}{4}

Hence, option B is the correct answer.

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