What `int sec x^(@) dx` is equal to ?

Options

A. `log (sec x^(@) + tan x^(@)) + c`
B. `(pi log tan ((pi)/(4) + (pi)/(2)))/(180^(@)) + c`
C. `(180^(@) log tan ((pi)/(4) + (x)/(2)))/(pi) + c`
D. `(180^(@) log tan ((pi)/(4) + (x)/(360^(@))))/(pi) + c`

Explanation

To solve the integral \(\int \sec(x^\circ) \, dx\), we need to convert the angle from degrees to radians. Here are the steps to solve the integral:

### Step-by-Step Solution:

1. **Convert Degrees to Radians**:
We know that \(1 \text{ degree} = \frac{\pi}{180} \text{ radians}\). Therefore, we can express \(x^\circ\) in radians as:
\[ x^\circ = \frac{\pi x}{180} \text{ radians} \]

2. **Substitute into the Integral**:
We can rewrite the integral as:
\[ \int \sec\left(\frac{\pi x}{180}\right) \, dx \]

3. **Change of Variable**:
Let \(t = \frac{\pi x}{180}\). Then, we differentiate to find \(dx\):
\[ dt = \frac{\pi}{180} \, dx \implies dx = \frac{180}{\pi} \, dt \]

4. **Substituting \(dx\) in the Integral**:
Now substitute \(dx\) in the integral:
\[ \int \sec(t) \cdot \frac{180}{\pi} \, dt = \frac{180}{\pi} \int \sec(t) \, dt \]

5. **Integrate \(\sec(t)\)**:
The integral of \(\sec(t)\) is:
\[ \int \sec(t) \, dt = \ln |\sec(t) + \tan(t)| + C \]
Therefore, we have:
\[ \frac{180}{\pi} \left(\ln |\sec(t) + \tan(t)| + C\right) = \frac{180}{\pi} \ln |\sec(t) + \tan(t)| + C' \]
where \(C' = \frac{180}{\pi} C\).

6. **Back Substitute for \(t\)**:
Recall that \(t = \frac{\pi x}{180}\). Substitute back:
\[ \int \sec(x^\circ) \, dx = \frac{180}{\pi} \ln \left|\sec\left(\frac{\pi x}{180}\right) + \tan\left(\frac{\pi x}{180}\right)\right| + C' \]

7. **Final Result**:
Thus, the final result for the integral \(\int \sec(x^\circ) \, dx\) is:
\[ \int \sec(x^\circ) \, dx = \frac{180}{\pi} \ln \left|\sec\left(\frac{\pi x}{180}\right) + \tan\left(\frac{\pi x}{180}\right)\right| + C \]