`int_0^(pi//2) (sin^(1//2)x)/(sin^(1//2)x+cos^(1//2)x) dx` is equal to

Options

A. 0
B. `pi/2`
C. `pi/3`
D. `pi/4`

Answer

To solve the integral

\[ I = \int_0^{\frac{\pi}{2}} \frac{\sin^{\frac{1}{2}} x}{\sin^{\frac{1}{2}} x + \cos^{\frac{1}{2}} x} \, dx, \]

we can use a symmetry property of definite integrals.

### Step 1: Define the integral
Let

\[ I = \int_0^{\frac{\pi}{2}} \frac{\sin^{\frac{1}{2}} x}{\sin^{\frac{1}{2}} x + \cos^{\frac{1}{2}} x} \, dx. \]

### Step 2: Change of variable
Now, we will perform a change of variable. Let \( x = \frac{\pi}{2} - t \). Then, \( dx = -dt \). The limits change as follows:
- When \( x = 0 \), \( t = \frac{\pi}{2} \)
- When \( x = \frac{\pi}{2} \), \( t = 0 \)

Thus, we have:

\[ I = \int_{\frac{\pi}{2}}^0 \frac{\sin^{\frac{1}{2}}(\frac{\pi}{2} - t)}{\sin^{\frac{1}{2}}(\frac{\pi}{2} - t) + \cos^{\frac{1}{2}}(\frac{\pi}{2} - t)} (-dt). \]

### Step 3: Simplify the integrand
Using the identities \( \sin(\frac{\pi}{2} - t) = \cos t \) and \( \cos(\frac{\pi}{2} - t) = \sin t \), we can rewrite the integral:

\[ I = \int_0^{\frac{\pi}{2}} \frac{\cos^{\frac{1}{2}} t}{\cos^{\frac{1}{2}} t + \sin^{\frac{1}{2}} t} \, dt. \]

### Step 4: Combine the two integrals
Now we have two expressions for \( I \):

1. \( I = \int_0^{\frac{\pi}{2}} \frac{\sin^{\frac{1}{2}} x}{\sin^{\frac{1}{2}} x + \cos^{\frac{1}{2}} x} \, dx \)
2. \( I = \int_0^{\frac{\pi}{2}} \frac{\cos^{\frac{1}{2}} x}{\sin^{\frac{1}{2}} x + \cos^{\frac{1}{2}} x} \, dx \)

Adding these two integrals, we get:

\[ 2I = \int_0^{\frac{\pi}{2}} \left( \frac{\sin^{\frac{1}{2}} x}{\sin^{\frac{1}{2}} x + \cos^{\frac{1}{2}} x} + \frac{\cos^{\frac{1}{2}} x}{\sin^{\frac{1}{2}} x + \cos^{\frac{1}{2}} x} \right) dx. \]

### Step 5: Simplify the combined integral
The expression simplifies to:

\[ 2I = \int_0^{\frac{\pi}{2}} \frac{\sin^{\frac{1}{2}} x + \cos^{\frac{1}{2}} x}{\sin^{\frac{1}{2}} x + \cos^{\frac{1}{2}} x} \, dx = \int_0^{\frac{\pi}{2}} 1 \, dx. \]

### Step 6: Evaluate the integral
The integral evaluates to:

\[ 2I = \left[ x \right]_0^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2}. \]

### Step 7: Solve for \( I \)
Dividing both sides by 2 gives:

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

### Final Answer
Thus, the value of the integral is

\[ \int_0^{\frac{\pi}{2}} \frac{\sin^{\frac{1}{2}} x}{\sin^{\frac{1}{2}} x + \cos^{\frac{1}{2}} x} \, dx = \frac{\pi}{4}. \]