integrate xlnx
Understand the Problem
The question is asking for the integral of the function xln(x) with respect to x. This involves applying integration techniques, possibly using integration by parts.
Answer
The integral is $$ \frac{x^2 \ln(x)}{2}  \frac{x^2}{4} + C $$
Answer for screen readers
The integral of $x \ln(x)$ with respect to $x$ is:
$$ \int x \ln(x) , dx = \frac{x^2 \ln(x)}{2}  \frac{x^2}{4} + C $$
Steps to Solve

Identify the function and the method
The function we want to integrate is $x \ln(x)$. Integration by parts is appropriate here, where we can let: $u = \ln(x)$ (which will be differentiated)
 $dv = x , dx$ (which will be integrated)

Differentiate and integrate
Now, differentiate $u$ and integrate $dv$: $du = \frac{1}{x} , dx$
 $v = \frac{x^2}{2}$

Apply the integration by parts formula
Using the integration by parts formula, $\int u , dv = uv  \int v , du$:
$$ \int x \ln(x) , dx = \ln(x) \cdot \frac{x^2}{2}  \int \frac{x^2}{2} \cdot \frac{1}{x} , dx $$
This simplifies to:
$$ = \frac{x^2 \ln(x)}{2}  \int \frac{x}{2} , dx $$ 
Integrate the remaining integral
Now we integrate the remaining expression:
$$ \int \frac{x}{2} , dx = \frac{1}{2} \cdot \frac{x^2}{2} = \frac{x^2}{4} $$ 
Put it all together
Now substituting back into our equation gives:
$$ \int x \ln(x) , dx = \frac{x^2 \ln(x)}{2}  \frac{x^2}{4} + C $$
where $C$ is the constant of integration.
The integral of $x \ln(x)$ with respect to $x$ is:
$$ \int x \ln(x) , dx = \frac{x^2 \ln(x)}{2}  \frac{x^2}{4} + C $$
More Information
This result uses the technique of integration by parts, which is often employed for products of polynomial and logarithmic functions. It exemplifies how complex integration can be broken down into manageable steps using strategic substitutions.
Tips
 Forgetting to include the constant of integration $C$ at the end.
 Misapplying the integration by parts formula. Always ensure you clearly identify $u$ and $dv$.