`I=int_(pi//5)^(3pi//10) (sinx)/(sinx+cosx)dx`is equal to

Options

A. `(pi)`
B. `(pi)/(2)`
C. `(pi)/(4)`
D. none of these

Explanation

To solve the integral

\[ I = \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{\sin x}{\sin x + \cos x} \, dx, \]

we will use a property of definite integrals. This property states that:

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

### Step 1: Identify \( a \) and \( b \)

Here, we have \( a = \frac{\pi}{5} \) and \( b = \frac{3\pi}{10} \). First, we need to calculate \( a + b \):

\[ a + b = \frac{\pi}{5} + \frac{3\pi}{10} = \frac{2\pi}{10} + \frac{3\pi}{10} = \frac{5\pi}{10} = \frac{\pi}{2}. \]

### Step 2: Apply the property

Now, we can apply the property:

\[ I = \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{\sin x}{\sin x + \cos x} \, dx = \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{\sin\left(\frac{\pi}{2} - x\right)}{\sin\left(\frac{\pi}{2} - x\right) + \cos\left(\frac{\pi}{2} - x\right)} \, dx. \]

### Step 3: 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 rewrite the integral:

\[ I = \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{\cos x}{\cos x + \sin x} \, dx. \]

### Step 4: Add the two integrals

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

1. \( I = \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{\sin x}{\sin x + \cos x} \, dx \)
2. \( I = \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{\cos x}{\cos x + \sin x} \, dx \)

Adding these two equations:

\[ 2I = \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \left( \frac{\sin x + \cos x}{\sin x + \cos x} \right) \, dx = \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} 1 \, dx. \]

### Step 5: Calculate the integral of 1

The integral of 1 over the interval from \( \frac{\pi}{5} \) to \( \frac{3\pi}{10} \) is simply the length of the interval:

\[ 2I = \left[ x \right]_{\frac{\pi}{5}}^{\frac{3\pi}{10}} = \frac{3\pi}{10} - \frac{\pi}{5} = \frac{3\pi}{10} - \frac{2\pi}{10} = \frac{\pi}{10}. \]

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

Now, we can solve for \( I \):

\[ 2I = \frac{\pi}{10} \implies I = \frac{\pi}{20}. \]

### Final Answer

Thus, the value of the integral is

\[ \boxed{\frac{\pi}{20}}. \]