ज्ञात करें `int log x dx`

Answer

To solve the integral \( \int \log x \, dx \), we will use the method of integration by parts.

### Step-by-Step Solution:

1. **Identify Functions for Integration by Parts**:
We will use the formula for integration by parts:
\[ \int u \, dv = uv - \int v \, du \]
Here, we choose:
- \( u = \log x \) (which we will differentiate)
- \( dv = dx \) (which we will integrate)

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

3. **Apply Integration by Parts**:
Substitute \( u \), \( v \), \( du \), and \( dv \) into the integration by parts formula:
\[ \int \log x \, dx = x \log x - \int x \cdot \frac{1}{x} \, dx \]
Simplifying the integral:
\[ \int \log x \, dx = x \log x - \int 1 \, dx \]

4. **Integrate the Remaining Integral**:
The integral of \( 1 \) is:
\[ \int 1 \, dx = x \]
Thus, we have:
\[ \int \log x \, dx = x \log x - x + C \]
where \( C \) is the constant of integration.

5. **Final Result**:
Therefore, the final result is:
\[ \int \log x \, dx = x \log x - x + C \]