`int(sinx+cosx)/(sin(x+alpha))dx=`

Options

A. `x(cosalpha-sinalpha)+(cosalpha+sinalpha)log|sin(x-alpha)|+c`
B. `x(cosalpha+sinalpha)+(cosalpha-sinalpha)log|sin(x-alpha)|+c`
C. `x(cosalpha+sinalpha)+(cosalpha-sinalpha)log|sin(x+alpha)|+c`
D. `x(cosalpha+sinalpha)-(cosalpha-sinalpha)log|sin(x+alpha)|+c`

Answer

To solve the integral

\[ I = \int \frac{\sin x + \cos x}{\sin(x + \alpha)} \, dx, \]

we will break it down into two parts:

\[ I = I_1 + I_2, \]

where

\[ I_1 = \int \frac{\sin x}{\sin(x + \alpha)} \, dx \quad \text{and} \quad I_2 = \int \frac{\cos x}{\sin(x + \alpha)} \, dx. \]

### Step 1: Rewrite the Integrals

We can express \( \sin(x + \alpha) \) using the angle addition formula:

\[ \sin(x + \alpha) = \sin x \cos \alpha + \cos x \sin \alpha. \]

Thus, we rewrite \( I_1 \) and \( I_2 \):

\[ I_1 = \int \frac{\sin x}{\sin x \cos \alpha + \cos x \sin \alpha} \, dx, \]

\[ I_2 = \int \frac{\cos x}{\sin x \cos \alpha + \cos x \sin \alpha} \, dx. \]

### Step 2: Use Trigonometric Identities

Now, we can simplify \( I_1 \) and \( I_2 \) further. We can rewrite the integrals as follows:

1. For \( I_1 \):

\[ I_1 = \int \frac{\sin x}{\sin x \cos \alpha + \cos x \sin \alpha} \, dx = \int \frac{\sin x}{\sin x \cos \alpha + \cos x \sin \alpha} \cdot \frac{1}{\sin x} \, dx = \int \frac{1}{\cos \alpha + \cot x \sin \alpha} \, dx. \]

2. For \( I_2 \):

\[ I_2 = \int \frac{\cos x}{\sin x \cos \alpha + \cos x \sin \alpha} \, dx = \int \frac{\cos x}{\sin x \cos \alpha + \cos x \sin \alpha} \cdot \frac{1}{\cos x} \, dx = \int \frac{1}{\tan x \cos \alpha + \sin \alpha} \, dx. \]

### Step 3: Solve the Integrals

Now, we can integrate each part separately.

1. For \( I_1 \):

Using the substitution \( u = \tan x \), we have \( du = \sec^2 x \, dx \) or \( dx = \frac{du}{1 + u^2} \).

Thus,

\[ I_1 = \int \frac{1}{\cos \alpha + u \sin \alpha} \cdot \frac{du}{1 + u^2}. \]

This integral can be solved using standard integral formulas.

2. For \( I_2 \):

Similarly, we can solve \( I_2 \) using the same substitution.

### Step 4: Combine the Results

After solving both integrals, we combine the results:

\[ I = I_1 + I_2. \]

### Final Result

The final result will be expressed in terms of \( x \) and \( \alpha \), including logarithmic terms and constants.