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

Options

A. 0
B. `(a+b) (pi)/(2)`
C. `a+b`
D. `(a+b) (pi)/(4)`

Answer

To solve the integral

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

we will use a property of definite integrals and symmetry.

### Step 1: Define the integral
Let

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

### Step 2: Use the property of definite integrals
We will apply the property

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

In our case, we have:

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

### Step 3: Simplify using trigonometric identities
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:

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

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

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

Adding these two equations, we get:

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

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

\[ (a \sin x + b \cos x) + (a \cos x + b \sin x) = (a + b)(\sin x + \cos x). \]

Thus, we have:

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

### Step 6: Cancel the common terms
The \(\sin x + \cos x\) terms cancel out:

\[ 2I = (a + b) \int_{0}^{\frac{\pi}{2}} 1 \, dx. \]

### Step 7: Evaluate the integral
The integral of 1 from 0 to \(\frac{\pi}{2}\) is:

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

### Step 8: Solve for \( I \)
Substituting this back, we get:

\[ 2I = (a + b) \cdot \frac{\pi}{2}. \]

Dividing both sides by 2 gives:

\[ I = \frac{(a + b) \cdot \frac{\pi}{2}}{2} = \frac{(a + b) \cdot \pi}{4}. \]

### Final Answer
Thus, the value of the integral is:

\[ I = \frac{(a + b) \pi}{4}. \]