Problems on Techniques of Integration

Use the Integration by Parts. Set

u &=& \ln(x)\\
dv &=& x dx\;.


du &=&\displaystyle \frac{1}{x} dx\\
v &=& \displaystyle \frac{x^2}{2}\;.


\begin{displaymath}\int x \ln(x) dx = \frac{x^2}{2} \ln(x) - \int \frac{x^2}{2} \frac{1}{x} dx = \frac{x^2}{2} \ln(x) - \int \frac{x}{2}dx\end{displaymath}


\begin{displaymath}\int x \ln(x) dx = \frac{x^2}{2} \ln(x) - \frac{x^2}{4} + C\;.\end{displaymath}

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