If `int_(pi//6)^(pi//3)(sqrt(sinx))/(sqrt(cosx)+sqrt(sinx))dx=(k)/(4)`, then value of k equals

Options

A. `(pi)/(12)`
B. `(pi)/(3)`
C. `(pi)/(2)`
D. None of these

Answer

To solve the integral

\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\sin x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx, \]

we can 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, \( a = \frac{\pi}{6} \) and \( b = \frac{\pi}{3} \). Therefore, \( a + b = \frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{6} + \frac{2\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2} \).

Now, we can rewrite the integral:

\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\sin\left(\frac{\pi}{2} - x\right)}}{\sqrt{\cos\left(\frac{\pi}{2} - x\right)} + \sqrt{\sin\left(\frac{\pi}{2} - x\right)}} \, dx. \]

### Step 2: Simplify the integrand

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

\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx. \]

### Step 3: Combine the two integrals

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

1. \( I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\sin x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx \)
2. \( I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \)

Adding these two equations gives:

\[ 2I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \left( \frac{\sqrt{\sin x} + \sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} \right) \, dx = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} 1 \, dx. \]

### Step 4: Evaluate the integral

Now we can evaluate the integral:

\[ \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} 1 \, dx = \left[ x \right]_{\frac{\pi}{6}}^{\frac{\pi}{3}} = \frac{\pi}{3} - \frac{\pi}{6} = \frac{\pi}{6}. \]

Thus, we have:

\[ 2I = \frac{\pi}{6} \implies I = \frac{\pi}{12}. \]

### Step 5: Relate to \( k \)

According to the problem statement, we have:

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

Substituting \( I = \frac{\pi}{12} \):

\[ \frac{\pi}{12} = \frac{k}{4}. \]

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

Multiplying both sides by 4 gives:

\[ k = \frac{\pi}{12} \times 4 = \frac{\pi}{3}. \]

### Final Answer

Thus, the value of \( k \) is

\[ \boxed{\frac{\pi}{3}}. \]