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

Options

A. `xcos(a-b)+sin(b-a).log[sec(x-b)]+c`
B. `xsin(a-b)+cos(a-b).log[sec(x-b)]+c`
C. `xcos(a-b)+sin(a-b).log[sec(x-b)]+c`
D. `xcos(b-a)+sin(b-a).log[sec(x-b)]+c`

Answer

To solve the integral \( I = \int \frac{\cos(x-a)}{\cos(x-b)} \, dx \), we will follow the steps outlined in the video transcript.

### Step-by-Step Solution:

1. **Rewrite the Numerator**:
We start with the integral:
\[ I = \int \frac{\cos(x-a)}{\cos(x-b)} \, dx \]
We can rewrite \(\cos(x-a)\) using the angle addition formula:
\[ \cos(x-a) = \cos(x-b + (b-a)) = \cos(x-b)\cos(b-a) + \sin(x-b)\sin(b-a) \]
Thus, we have:
\[ I = \int \frac{\cos(x-b)\cos(b-a) + \sin(x-b)\sin(b-a)}{\cos(x-b)} \, dx \]

2. **Split the Integral**:
We can split the integral into two parts:
\[ I = \int \cos(b-a) \, dx + \int \frac{\sin(x-b)\sin(b-a)}{\cos(x-b)} \, dx \]
This simplifies to:
\[ I = \cos(b-a) \int dx + \sin(b-a) \int \tan(x-b) \, dx \]

3. **Integrate Each Part**:
- The first integral is straightforward:
\[ \int dx = x \]
- The second integral involves the tangent function:
\[ \int \tan(x-b) \, dx = -\log|\cos(x-b)| + C \]

4. **Combine the Results**:
Now, we can combine the results of the two integrals:
\[ I = \cos(b-a) x + \sin(b-a) (-\log|\cos(x-b)|) + C \]
This can be rewritten as:
\[ I = x \cos(b-a) - \sin(b-a) \log|\cos(x-b)| + C \]

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