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Question: Answered & Verified by Expert
$\int x \log x d x$ is equal to
MathematicsIndefinite IntegrationMHT CETMHT CET 2009
Options:
  • A $\frac{x^{2}}{4}(2 \log x-1)+c$
  • B $\frac{x^{2}}{2}(2 \log x-1)+c$
  • C $\frac{x^{2}}{4}(2 \log x+1)+c$
  • D $\frac{x^{2}}{2}(2 \log x+1)$
Solution:
2701 Upvotes Verified Answer
The correct answer is: $\frac{x^{2}}{4}(2 \log x-1)+c$
$\begin{aligned} \int \frac{x}{\Pi} \log _{I} x d x &=\log x \cdot \frac{x^{2}}{2}-\int \frac{1}{x} \cdot \frac{x^{2}}{2} d x \\ &=\frac{x^{2}}{2} \log x-\frac{1}{2} \frac{x^{2}}{2}+c \\ &=\frac{x^{2}}{4}(2 \log x-1)+c \end{aligned}$

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