Choose the correct Answer of the Following Questions : `int_0^(pi/2) sqrtcosx/(sqrtcosx + sqrtsinx) dx =`

Options

A. 0
B. `pi/2`
C. `pi/4`
D. `2pi`

Answer

To solve the definite integral \( I = \int_0^{\frac{\pi}{2}} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx \), we can use a symmetry property of definite integrals. Here’s the step-by-step solution:

### Step 1: Define the integral
Let
\[ I = \int_0^{\frac{\pi}{2}} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx. \]

### Step 2: Use the substitution \( x = \frac{\pi}{2} - t \)
We will apply the substitution \( x = \frac{\pi}{2} - t \). Then, \( dx = -dt \) and the limits change as follows:
- When \( x = 0 \), \( t = \frac{\pi}{2} \).
- When \( x = \frac{\pi}{2} \), \( t = 0 \).

Thus, we have:
\[ I = \int_{\frac{\pi}{2}}^0 \frac{\sqrt{\cos\left(\frac{\pi}{2} - t\right)}}{\sqrt{\cos\left(\frac{\pi}{2} - t\right)} + \sqrt{\sin\left(\frac{\pi}{2} - t\right)}} (-dt). \]

### Step 3: Simplify the integral
Using the trigonometric identities \( \cos\left(\frac{\pi}{2} - t\right) = \sin t \) and \( \sin\left(\frac{\pi}{2} - t\right) = \cos t \), we can rewrite the integral:
\[ I = \int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin t}}{\sqrt{\sin t} + \sqrt{\cos t}} \, dt. \]

### Step 4: Combine the two integrals
Now we have two expressions for \( I \):
1. \( I = \int_0^{\frac{\pi}{2}} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx \)
2. \( I = \int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \)

Adding these two equations gives:
\[ 2I = \int_0^{\frac{\pi}{2}} \left( \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} + \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \right) dx. \]

### Step 5: Simplify the combined integral
The denominator is the same for both fractions:
\[ 2I = \int_0^{\frac{\pi}{2}} \frac{\sqrt{\cos x} + \sqrt{\sin x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx = \int_0^{\frac{\pi}{2}} 1 \, dx. \]

### Step 6: Evaluate the integral
The integral \( \int_0^{\frac{\pi}{2}} 1 \, dx \) evaluates to:
\[ \int_0^{\frac{\pi}{2}} 1 \, dx = \left[ x \right]_0^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2}. \]

### Step 7: Solve for \( I \)
Now we have:
\[ 2I = \frac{\pi}{2} \implies I = \frac{\pi}{4}. \]

### Conclusion
Thus, the value of the integral is:
\[ \int_0^{\frac{\pi}{2}} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx = \frac{\pi}{4}. \]

### Final Answer
The correct answer is \( \frac{\pi}{4} \).

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