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

Options

A. `(1)/(cos(b-a))log|cos(x-a)/sin(x-b)|+c`
B. `(1)/(cos(b-a))log|sin(x-b)/cos(x-a)|+c`
C. `(1)/(sin(b-a))log|cos(x-a)/sin(x-b)|+c`
D. `(1)/(sin(b-a))log|sin(x-b)/cos(x-a)|+c`

Answer

To solve the integral

\[ I = \int \frac{dx}{\cos(x-a) \sin(x-b)}, \]

we can use a trigonometric identity and properties of logarithms. Let's go through the steps:

### Step 1: Rewrite the Integral
We start with the integral:

\[ I = \int \frac{dx}{\cos(x-a) \sin(x-b)}. \]

### Step 2: Use Trigonometric Identity
We can express the integrand using the identity for the difference of angles. We rewrite the integral as follows:

\[ I = \frac{1}{\cos(b-a)} \int \frac{\cos(b-a)}{\cos(x-a) \sin(x-b)} \, dx. \]

### Step 3: Apply the Cosine Difference Identity
Using the cosine difference identity, we have:

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

Substituting this into the integral gives us:

\[ I = \frac{1}{\cos(b-a)} \int \frac{\cos(b-a)}{\cos(x-a) \sin(x-b)} \, dx. \]

### Step 4: Separate the Integral
Now we can separate the integral into two parts:

\[ I = \frac{1}{\cos(b-a)} \left( \int \frac{\cos(x-b)}{\sin(x-b)} \, dx + \int \frac{\sin(x-a)}{\cos(x-a)} \, dx \right). \]

### Step 5: Use Logarithmic Properties
The integrals can be solved using logarithmic functions. The integral of \(\cot\) and \(\tan\) gives us:

\[ \int \cot(x-b) \, dx = \log|\sin(x-b)| + C_1, \]
\[ \int \tan(x-a) \, dx = -\log|\cos(x-a)| + C_2. \]

### Step 6: Combine the Results
Combining these results, we get:

\[ I = \frac{1}{\cos(b-a)} \left( \log|\sin(x-b)| - \log|\cos(x-a)| \right) + C. \]

### Step 7: Final Simplification
Using the properties of logarithms, we can combine the logs:

\[ I = \frac{1}{\cos(b-a)} \log\left|\frac{\sin(x-b)}{\cos(x-a)}\right| + C. \]

### Final Answer
Thus, the final result for the integral is:

\[ I = \frac{1}{\cos(b-a)} \log\left|\frac{\sin(x-b)}{\cos(x-a)}\right| + C. \]

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