Question

Evaluate the integral
${\int }_{1/2}^2|{\log }_{10}x|dx$

• A. ${\log }_{10}\left(8/e\right)$
• B. $\frac{1}{2}{\log }_{10}\left(8/e\right)$
• C. ${\log }_{10}\left(2/e\right)$
• D. ${\log }_e\left(3/e\right)$

Answer 1

The correct option is B $\frac{1}{2}{\log }_{10}\left(8/e\right)$

$I={\int }_{1/2}^2\left|{\log }_{10}x\right|dx={\int }_{1/2}^1\left|{\log }_{10}x\right|dx+{\int }_1^2\left|{\log }_{10}x\right|dx$
For $\frac{1}{2}<x<1\Rightarrow {\log }_{10}x<0$ and $1<x<2\Rightarrow {\log }_{10}x>0$
$I=-{\int }_{1/2}^1{\log }_{10}xdx+{\int }_1^2{\log }_{10}xdx=-{\left[\frac{x(\log x-1)}{\log 10}\right]}_{1/2}^1+{\left[\frac{x(\log x-1)}{\log 10}\right]}_1^2=\frac{1}{2}{\log }_{10}\frac{8}{e}$