`int x log x, dx`

Options

A. `(1)/(4)(2x^(2)logx-1)+c`
B. `(x^(2))/(4)(2logx+1)+c`
C. `(x^(2))/(2)(2logx-1)+c`
D. `(x^(2))/(4)(2logx-1)+c`

Answer

To solve the integral \( \int x \log x \, dx \), we will use the method of integration by parts. The formula for integration by parts is given by:

\[ \int u \, dv = uv - \int v \, du \]

### Step 1: Choose \( u \) and \( dv \)

Let:
- \( u = \log x \) (which we will differentiate)
- \( dv = x \, dx \) (which we will integrate)

### Step 2: Differentiate \( u \) and Integrate \( dv \)

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

### Step 3: Apply the Integration by Parts Formula

Now, we can apply the integration by parts formula:
\[ \int x \log x \, dx = uv - \int v \, du \]
Substituting the values of \( u \), \( v \), \( du \):
\[ \int x \log x \, dx = \log x \cdot \frac{x^2}{2} - \int \frac{x^2}{2} \cdot \frac{1}{x} \, dx \]

### Step 4: Simplify the Integral

The integral simplifies as follows:
\[ \int x \log x \, dx = \frac{x^2 \log x}{2} - \int \frac{x^2}{2x} \, dx \]
This simplifies to:
\[ \int x \log x \, dx = \frac{x^2 \log x}{2} - \int \frac{x}{2} \, dx \]

### Step 5: Solve the Remaining Integral

Now we need to solve \( \int \frac{x}{2} \, dx \):
\[ \int \frac{x}{2} \, dx = \frac{1}{2} \cdot \frac{x^2}{2} = \frac{x^2}{4} \]

### Step 6: Combine the Results

Putting it all together, we have:
\[ \int x \log x \, dx = \frac{x^2 \log x}{2} - \frac{x^2}{4} + C \]

### Final Answer

Thus, the final result is:
\[ \int x \log x \, dx = \frac{x^2 \log x}{2} - \frac{x^2}{4} + C \]