`intx^nlogxdx =`

Options

A. `(x^(n+1))/n[n log x -1]`
B. `(x^(n+1))/(n+1)[(n+1)logx-1]`
C. `(x^(n+1))/((n+1)^2)[(n+1)logx-1]`
D. none

Answer

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

### Step-by-Step Solution:

1. **Identify the parts for integration by parts**:
We choose:
- \( u = \log x \) (which we will differentiate)
- \( dv = x^n \, dx \) (which we will integrate)

2. **Differentiate and integrate**:
- Differentiate \( u \):
\[ du = \frac{1}{x} \, dx \]
- Integrate \( dv \):
\[ v = \int x^n \, dx = \frac{x^{n+1}}{n+1} \]

3. **Apply the integration by parts formula**:
The formula for integration by parts is:
\[ \int u \, dv = uv - \int v \, du \]
Substituting our values:
\[ \int x^n \log x \, dx = \left( \log x \cdot \frac{x^{n+1}}{n+1} \right) - \int \left( \frac{x^{n+1}}{n+1} \cdot \frac{1}{x} \right) \, dx \]

4. **Simplify the integral**:
The integral simplifies to:
\[ \int x^n \log x \, dx = \frac{x^{n+1} \log x}{n+1} - \frac{1}{n+1} \int x^n \, dx \]

5. **Integrate \( \int x^n \, dx \)**:
We know:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} \]
Substituting this back into our equation:
\[ \int x^n \log x \, dx = \frac{x^{n+1} \log x}{n+1} - \frac{1}{n+1} \cdot \frac{x^{n+1}}{n+1} \]

6. **Combine the terms**:
\[ \int x^n \log x \, dx = \frac{x^{n+1} \log x}{n+1} - \frac{x^{n+1}}{(n+1)^2} \]
Factor out \( \frac{x^{n+1}}{n+1} \):
\[ \int x^n \log x \, dx = \frac{x^{n+1}}{n+1} \left( \log x - \frac{1}{n+1} \right) + C \]

### Final Answer:
\[ \int x^n \log x \, dx = \frac{x^{n+1}}{n+1} \left( \log x - \frac{1}{n+1} \right) + C \]