Q250 / 270 · NDA 2017 (Phase 1) · Mathematics

What is $\int_{\mathrm{e}^{-1}}^{\mathrm{e}^2}\left|\frac{\ln \mathrm{x}}{\mathrm{x}}\right| \mathrm{dx}$ equal to?

  • A
    3/2
  • Correct answer
    B
    5/2
  • C
    3
  • D
    4

Solution

$\begin{aligned} & \int_{e^{-1}}^{e^2}\left|\frac{\log x}{x}\right| d x=\int_{e^{-1}}^{e^0} \frac{-\log x}{x} d x+\int_{e^0}^{e^2} \frac{\log x}{x} d x \\ & =\frac{-1}{2}\left[(\log x)^2\right]_{e^{-1}}^{e^0}+\frac{1}{2}\left[(\log x)^2\right]_{e^0}^{e^2} \\ & =\frac{-1}{2}\left[0-\left(\log e^{-1}\right)^2\right]+\frac{1}{2}\left[\left(\log e^2\right)^2-0\right] \\ & =\frac{-1}{2}(-1)+\frac{1}{2}(2)^2=\frac{1}{2}+\frac{4}{2}=\frac{5}{2}\end{aligned}$

Hence, the correct answer is option: 'B'.

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