Q. $\int_{}^{}{x\log x}\ dx$ is equal to
A) $\frac{x^{2}}{4}\ \left( 2\log{x - 1} \right) + c$
B) $\frac{x^{2}}{2}\ \left( 2\log{x - 1} \right) + c$
C) $\frac{x^{2}}{4}\ \left( 2\log{x + 1} \right) + c$
D) $\frac{x^{2}}{2}\ \left( 2\log{x + 1} \right) + c$
Explanation
(a)
$\int_{}^{}{\begin{matrix}
x \\
II
\end{matrix}\begin{matrix}
\log x \\
I
\end{matrix}}\ dx = \log x.\frac{x^{2}}{2} - \int_{}^{}\frac{1}{x}.\frac{x^{2}}{2}\ dx$
$= \frac{x^{2}}{2}\log x - \frac{1}{2}\ \frac{x^{2}}{2} + c$
$= \frac{x^{2}}{4}\left( 2\log x - 1 \right) + c$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.