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UPSC NDA Math Model Paper 1

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Question : 1 of 120

Marks: +1, -0

The value of the integral

; \int\limits_{0}^{\pi / 2} \; \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} \, dx

\int\limits_{0}^{\pi / 2} \; \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} \, dx \; \text{ is } \;

0
$-\pi/4-\pi/4$
$\pi/2 \pi/2$
$\pi/4 \pi/4$

Solution:

\;\text{ Concept: }\;

\int\limits_{a}^{b} f(x) \, dx

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

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

Calculations:

Consider,

I = \int\limits_{0}^{\pi / 2} \; \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} \, dx

\int\limits_{0}^{\pi / 2} \; \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} \, dx

I = \int\limits_{0}^{\pi / 2} \frac{\sqrt{\sin\left(\frac{\pi}{2} - x\right)}}{\sqrt{\sin\left(\frac{\pi}{2} - x\right)} + \sqrt{\cos\left(\frac{\pi}{2} - x\right)}} \, dx

\int\limits_{0}^{\pi / 2} \frac{\sqrt{\sin\left(\frac{\pi}{2} - x\right)}}{\sqrt{\sin\left(\frac{\pi}{2} - x\right)} + \sqrt{\cos\left(\frac{\pi}{2} - x\right)}} \, dx

I = \int\limits_{0}^{\pi / 2} \; \frac{\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}} \, dx

\int\limits_{0}^{\pi / 2} \; \frac{\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}} \, dx

Adding (1) and (2), we have

2 I = \int\limits_{0}^{\pi / 2} \; \frac{\sqrt{\cos x} + \sqrt{\sin x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx

\int\limits_{0}^{\pi / 2} \; \frac{\sqrt{\cos x} + \sqrt{\sin x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx

2 I = \int\limits_{0}^{\pi / 2} \, dx \int\limits_{0}^{\pi / 2} \, dx

2 I = [x]_{0}^{\;\frac{\pi}{2}} [x]^{\;\frac{\pi}{\theta}}

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

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