`int_(0)^(pi//2) sqrt(sin x)/(sqrt(sin x ) + sqrt( cos x)) dx = `

Options

A. `(3pi)/2`
B. `(3pi)/4`
C. `pi/2`
D. `pi/4`

Answer

To solve the integral

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

we can use the property of definite integrals, which states that

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

In our case, \( a = 0 \) and \( b = \frac{\pi}{2} \). Thus, we can rewrite the integral as follows:

1. **Substituting \( x \) with \( \frac{\pi}{2} - x \)**:

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

Using the identities \( \sin\left(\frac{\pi}{2} - x\right) = \cos x \) and \( \cos\left(\frac{\pi}{2} - x\right) = \sin x \), we can rewrite the integral:

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

2. **Adding the two expressions for \( I \)**:

Now we have two expressions for \( I \):

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

Adding these two equations gives:

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

3. **Simplifying the integrand**:

The integrand simplifies as follows:

\[ \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} + \frac{\sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} = \frac{\sqrt{\sin x} + \sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} = 1. \]

Thus, we have:

\[ 2I = \int_{0}^{\frac{\pi}{2}} 1 \, dx. \]

4. **Calculating the integral**:

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

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

5. **Solving for \( I \)**:

Now we can solve for \( I \):

\[ 2I = \frac{\pi}{2} \implies I = \frac{\pi}{4}. \]

Thus, the value of the integral is:

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

**Final Answer**:

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