First slide

Evaluation of definite integrals

Question

$\int_{e^{- 1}}^{e^{2}} | \frac{\ln x}{x} | d x =$

Moderate

A

$\frac{3}{2}$
B

$\frac{5}{2}$
C

3
D

5

Solution

$I = \int_{e^{- 1}}^{e^{2}} | \frac{\ln x}{x} | d x = \int_{e^{- 1}}^{1} - \frac{\ln x}{x} d x + \int_{1}^{e^{2}} \frac{\ln x}{x} d x$
$= - \frac{1}{2} {(\ln x)}^{2} {| \begin{matrix} 1 \\ e^{- 1} \end{matrix} + \frac{1}{2} {(\ln x)}^{2} |}_{1}^{e^{2}}$

$= \frac{1}{2} + \frac{1}{2} \times 4 = \frac{5}{2}$

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