`int_(0)^(pi//2)(sqrt(cosx))/((sqrt(cosx)+sqrt(sinx)))dx=?`

Options

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

Answer

To solve the integral

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

we will use the property of definite integrals that states:

\[ \int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx. \]

### Step 1: Apply the property of definite integrals

Here, we have \( a = 0 \) and \( b = \frac{\pi}{2} \). Thus, we can rewrite the integral as:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\cos\left(\frac{\pi}{2} - x\right)}}{\sqrt{\cos\left(\frac{\pi}{2} - x\right)} + \sqrt{\sin\left(\frac{\pi}{2} - x\right)}} \, dx. \]

### Step 2: Simplify the expression

Using the trigonometric identities, we know:

\[ \cos\left(\frac{\pi}{2} - x\right) = \sin x \quad \text{and} \quad \sin\left(\frac{\pi}{2} - x\right) = \cos x. \]

Thus, we can rewrite the integral as:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx. \]

### Step 3: 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, we get:

\[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} + \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \right) dx. \]

### Step 4: Simplify the sum

The denominators are the same, so we can combine the numerators:

\[ 2I = \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\cos x} + \sqrt{\sin x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx = \int_{0}^{\frac{\pi}{2}} 1 \, dx. \]

### Step 5: Evaluate the integral

The integral of 1 from 0 to \(\frac{\pi}{2}\) is simply:

\[ \int_{0}^{\frac{\pi}{2}} 1 \, dx = \left[ x \right]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2}. \]

### Step 6: Solve for \( I \)

Now we have:

\[ 2I = \frac{\pi}{2}. \]

Dividing both sides by 2 gives:

\[ I = \frac{\pi}{4}. \]

### Final Answer

Thus, the value of the integral is:

\[ \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx = \frac{\pi}{4}. \]

---