Q117 / 160 · AP EAMCET 2019 (20 Apr Shift 2) · Mathematics

$\int_{e^{-1}}^{e^2}\left|\frac{\log x}{x}\right| d x=$

  • A
    $\frac{2}{5}$
  • B
    2
  • C
    5
  • Correct answer
    D
    $\frac{5}{2}$

Solution

$I=\int_{e^{-1}}^{e^2}\left|\frac{\log x}{x}\right| d x=\int_{e^{-1}}^1\left(-\frac{\log x}{x}\right) d x+\int_1^{e^2}\left(\frac{\log x}{x}\right) d x$ On putting $\log x=t \Rightarrow \frac{d x}{x}=d t$ $$ \begin{aligned} & =-\int_{-1}^0 t d t+\int_0^2 t d t=\left[\frac{-t^2}{2}\right]_{-1}^0+\left[\frac{t^2}{2}\right]_0^2 \\ & =-\frac{1}{2}(0-1)+\frac{1}{2}(4-0)=\frac{1}{2}+2=\frac{5}{2} \end{aligned} $$ Hence, option (d) is correct.

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

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