The integral `intcos(log_(e)x)dx` is equal to: (where C is a constant of integration)

Options

A. `x/2[sin(log_ex-cos(log_ex)]+C`
B. `x/2[cos(log_ex+sin(log_ex)]+C`
C. `x[cos(log_ex+sin(log_ex)]+C`
D. `x[cos(log_ex-sin(log_ex)]+C`

Answer

To solve the integral \( \int \cos(\log x) \, dx \), we will use integration by parts. Let's go through the solution step by step.

### Step-by-Step Solution:

1. **Identify the integral**:
\[ I = \int \cos(\log x) \, dx \]

2. **Use integration by parts**:
We will use the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
Let's choose:
- \( u = \cos(\log x) \)
- \( dv = dx \)

3. **Differentiate and integrate**:
Now we need to find \( du \) and \( v \):
- Differentiate \( u \):
\[ du = -\sin(\log x) \cdot \frac{1}{x} \, dx \]
- Integrate \( dv \):
\[ v = x \]

4. **Apply the integration by parts formula**:
Substitute \( u \), \( du \), \( v \), and \( dv \) into the integration by parts formula:
\[ I = x \cos(\log x) - \int x \left(-\sin(\log x) \cdot \frac{1}{x}\right) \, dx \]
Simplifying gives:
\[ I = x \cos(\log x) + \int \sin(\log x) \, dx \]

5. **Evaluate the new integral**:
Let’s denote the new integral as \( J = \int \sin(\log x) \, dx \). We will again use integration by parts:
- Choose:
- \( u = \sin(\log x) \)
- \( dv = dx \)
- Then:
- \( du = \cos(\log x) \cdot \frac{1}{x} \, dx \)
- \( v = x \)

6. **Apply integration by parts again**:
\[ J = x \sin(\log x) - \int x \cos(\log x) \cdot \frac{1}{x} \, dx \]
Simplifying gives:
\[ J = x \sin(\log x) - \int \cos(\log x) \, dx \]
Notice that \( \int \cos(\log x) \, dx = I \), so we can substitute:
\[ J = x \sin(\log x) - I \]

7. **Combine the results**:
Now we substitute \( J \) back into the equation for \( I \):
\[ I = x \cos(\log x) + (x \sin(\log x) - I) \]
Rearranging gives:
\[ 2I = x \cos(\log x) + x \sin(\log x) \]
Therefore:
\[ I = \frac{1}{2} (x \cos(\log x) + x \sin(\log x)) \]

8. **Final result**:
The integral is:
\[ \int \cos(\log x) \, dx = \frac{x}{2} (\cos(\log x) + \sin(\log x)) + C \]
where \( C \) is the constant of integration.