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

Options

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

Answer

To solve the integral

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

we can use the property of definite integrals, specifically the substitution \( x = \frac{\pi}{2} - t \).

### Step 1: Set up the integral
Let

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

### Step 2: Apply the substitution
Using the substitution \( x = \frac{\pi}{2} - t \), we have \( dx = -dt \). When \( x = 0 \), \( t = \frac{\pi}{2} \) and when \( x = \frac{\pi}{2} \), \( t = 0 \). Thus, the limits of integration change, and we can rewrite the integral as:

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

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

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\cos t}{\cos t + \sin t} \, dt. \]

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

1. \( I = \int_{0}^{\frac{\pi}{2}} \frac{\sin x}{\sin x + \cos x} \, dx \)
2. \( I = \int_{0}^{\frac{\pi}{2}} \frac{\cos x}{\cos x + \sin x} \, dx \)

Adding these two integrals:

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

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

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

Thus, we have:

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

### Step 6: Evaluate the integral
Now, we can evaluate the integral:

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

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

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

### Conclusion
Thus, the value of the integral is

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