`int(sin(x+a))/(cos(x-b))dx=`

Options

A. `x sin(a+b)+cos(a+b)log|sec(x-b)|+c`
B. `x sin(a-b)+cos(a-b)log|sec(x-b)|+c`
C. `x sin(a-b)-cos(a-b)log|sec(x-b)|+c`
D. `x sin(a+b)-cos(a+b)log|sec(x-b)|+c`

Answer

To solve the integral \( I = \int \frac{\sin(x + a)}{\cos(x - b)} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the angle in the sine function. We can express \( x + a \) as:
\[ x + a = (x - b) + (a + b) \]
This allows us to rewrite the integral as:
\[ I = \int \frac{\sin((x - b) + (a + b))}{\cos(x - b)} \, dx \]

### Step 2: Use the Sine Addition Formula
Using the sine addition formula \( \sin(A + B) = \sin A \cos B + \cos A \sin B \), we can expand the sine function:
\[ \sin((x - b) + (a + b)) = \sin(x - b) \cos(a + b) + \cos(x - b) \sin(a + b) \]
Substituting this back into the integral gives:
\[ I = \int \frac{\sin(x - b) \cos(a + b) + \cos(x - b) \sin(a + b)}{\cos(x - b)} \, dx \]

### Step 3: Simplify the Integral
Now we can separate the integral:
\[ I = \int \left( \cos(a + b) \tan(x - b) + \sin(a + b) \right) \, dx \]

### Step 4: Integrate Each Term
Now we can integrate each term separately:
1. The integral of \( \tan(x - b) \):
\[ \int \tan(x - b) \, dx = -\log|\cos(x - b)| \]
2. The integral of a constant \( \sin(a + b) \):
\[ \int \sin(a + b) \, dx = \sin(a + b) x \]

Putting these together, we have:
\[ I = \cos(a + b) \left(-\log|\cos(x - b)|\right) + \sin(a + b) x + C \]

### Step 5: Final Result
Thus, the final result for the integral is:
\[ I = \sin(a + b) x - \cos(a + b) \log|\cos(x - b)| + C \]