xlogxの定積分

$$ \begin{aligned} \hspace{10pt}& \int_1^e x\log x \hspace{1pt}dx \\[1em] &= \int_1^e \left(\frac{1}{2}x^2 \right)' \log x \hspace{1pt}dx \\[1em] &= \left[\frac{1}{2}x^2 \log x \right]_1^e - \int_1^e \frac{1}{2}x^2 \cdot \frac{1}{x} \hspace{1pt}dx\\[1em] &= \frac{1}{2}e^2 - \frac{1}{2} \int_1^e x \hspace{1pt}dx\\[1em] &= \frac{1}{2}e^2 - \frac{1}{2} \left[ \frac{1}{2}x^2\right]_1^e\\[1em] &= \frac{1}{2}e^2 - \frac{1}{4} (e^2 -1)\\[1em] &= \frac{1}{4}(e^2 + 1) \\[1em] \end{aligned} $$