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 given problem, we need to evaluate the 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:

\[ \int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a - x) \, dx \]

In our case, let \( a = \frac{\pi}{2} \). Thus,

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin\left(\frac{\pi}{2} - x\right)}{1 + \sin\left(\frac{\pi}{2} - x\right) + \cos\left(\frac{\pi}{2} - x\right)} \, dx \]

### Step 2: Simplify the integral

Using the trigonometric identities, we have:

\[ \sin\left(\frac{\pi}{2} - x\right) = \cos x \quad \text{and} \quad \cos\left(\frac{\pi}{2} - x\right) = \sin x \]

Thus, we can rewrite the integral as:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\cos x}{1 + \cos x + \sin x} \, dx \]

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

Adding these two equations gives:

\[ 2I = \int_{0}^{\frac{\pi}{2}} \frac{\sin x + \cos x}{1 + \sin x + \cos x} \, dx \]

### Step 4: Simplify the numerator

Notice that:

\[ \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \]

However, we will keep it as \( \sin x + \cos x \) for simplicity. Thus:

\[ 2I = \int_{0}^{\frac{\pi}{2}} \frac{\sin x + \cos x}{1 + \sin x + \cos x} \, dx \]

### Step 5: Evaluate the integral

Now we can separate the integral:

\[ 2I = \int_{0}^{\frac{\pi}{2}} \frac{\sin x + \cos x}{1 + \sin x + \cos x} \, dx \]

This integral can be evaluated using the known value of \( A \):

\[ 2I = A \]

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

Thus, we find:

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

### Final Result

The value of the definite integral

\[ \int_{0}^{\frac{\pi}{2}} \frac{\sin x}{1 + \sin x + \cos x} \, dx = \frac{A}{2} \]