Given `int_(0)^(pi//2)(dx)/(1+sinx+cosx)=A`. Then the value of the definite integral `int_(0)^(pi//2)(sinx)/(1+sinx+cosx)dx` is equal to

Options

A. `1/2A`
B. `(pi)/2-A`
C. `(pi)/4-1/2A`
D. `(pi)/2+A`

Explanation

To solve the problem, we need to evaluate the definite integral

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

We are also given that

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

### Step 1: Use the property of definite integrals

We can use the property of definite integrals that states:

\[ \int_0^a f(x) \, dx = \int_0^a f(a - x) \, dx. \]

Let’s apply this property to our integral \(I\):

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

### Step 2: Simplify the integral

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

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

### Step 3: Add the two integrals

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

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

Adding these two integrals gives:

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

### Step 4: Rewrite the integral

We can separate the integral:

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

This can be rewritten as:

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

### Step 5: Substitute the known integral

From the problem statement, we know:

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

Thus, we can write:

\[ 2I = A. \]

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

Dividing both sides by 2, we find:

\[ I = \frac{A}{2}. \]

### Conclusion

The value of the definite integral

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