`int_(1)^(2)x log x dx =`

Options

A. `(3)/(4)-(log 4)`
B. `(log 4)-(3)/(4)`
C. `e^(2)-e`
D. `e+e^(2)`

Answer

To solve the definite integral \( \int_{1}^{2} x \log x \, dx \), we will use 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 implies \( du = \frac{1}{x} \, dx \))
- \( dv = x \, dx \) (which implies \( v = \frac{x^2}{2} \))

### Step 2: Apply integration by parts
Using the integration by parts formula, we have:

\[ \int x \log x \, dx = uv - \int v \, du \]

Substituting \( u \), \( du \), \( v \), and \( dv \):

\[ \int x \log x \, dx = \log x \cdot \frac{x^2}{2} - \int \frac{x^2}{2} \cdot \frac{1}{x} \, dx \]

This simplifies to:

\[ \int x \log x \, dx = \frac{x^2 \log x}{2} - \int \frac{x^2}{2x} \, dx \]

\[ = \frac{x^2 \log x}{2} - \int \frac{x}{2} \, dx \]

### Step 3: Solve the remaining integral
Now, we can solve the integral \( \int \frac{x}{2} \, dx \):

\[ \int \frac{x}{2} \, dx = \frac{1}{2} \cdot \frac{x^2}{2} = \frac{x^2}{4} \]

### Step 4: Combine the results
Substituting back, we have:

\[ \int x \log x \, dx = \frac{x^2 \log x}{2} - \frac{x^2}{4} \]

### Step 5: Evaluate the definite integral from 1 to 2
Now we need to evaluate:

\[ \int_{1}^{2} x \log x \, dx = \left[ \frac{x^2 \log x}{2} - \frac{x^2}{4} \right]_{1}^{2} \]

Calculating at the upper limit \( x = 2 \):

\[ = \frac{2^2 \log 2}{2} - \frac{2^2}{4} = \frac{4 \log 2}{2} - \frac{4}{4} = 2 \log 2 - 1 \]

Calculating at the lower limit \( x = 1 \):

\[ = \frac{1^2 \log 1}{2} - \frac{1^2}{4} = \frac{0}{2} - \frac{1}{4} = -\frac{1}{4} \]

### Step 6: Combine the results
Now, we combine the results from the upper and lower limits:

\[ \int_{1}^{2} x \log x \, dx = \left( 2 \log 2 - 1 \right) - \left( -\frac{1}{4} \right) \]

\[ = 2 \log 2 - 1 + \frac{1}{4} = 2 \log 2 - \frac{4}{4} + \frac{1}{4} = 2 \log 2 - \frac{3}{4} \]

### Final Result
Thus, the value of the definite integral is:

\[ \int_{1}^{2} x \log x \, dx = 2 \log 2 - \frac{3}{4} \]