NIMCET 2013 — Mathematics PYQ

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Question

NIMCET 2013 — Mathematics PYQ

NIMCET | Mathematics | 2013

The value of the integral $\int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$ is:

Choose the correct answer:

Explanation

Solution

Step 1: Assign the integral to a variable $I$

Let

I = \int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx \quad \text{--- (Equation 1)}

Step 2: Apply the property $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a + b - x) dx$

Here, $a = 0$ and $b = \pi/2$ . We replace $x$ with $(\pi/2 - x)$ :

I = \int_{0}^{\pi/2} \frac{\sqrt{\sin(\pi/2 - x)}}{\sqrt{\sin(\pi/2 - x)} + \sqrt{\cos(\pi/2 - x)}} dx

Using trigonometric identities $\sin(\pi/2 - x) = \cos x$ and $\cos(\pi/2 - x) = \sin x$ :

I = \int_{0}^{\pi/2} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} dx \quad \text{--- (Equation 2)}

Step 3: Add Equation 1 and Equation 2

I + I = \int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx + \int_{0}^{\pi/2} \frac{\sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx

2I = \int_{0}^{\pi/2} \frac{\sqrt{\sin x} + \sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx

Step 4: Simplify and integrate

2I = \int_{0}^{\pi/2} 1 \, dx

2I = [x]_{0}^{\pi/2}

2I = \frac{\pi}{2} - 0

I = \frac{\pi}{4}

Correct Option: 4. $\frac{\pi}{4}$

Choose the correct answer:

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