Question

${\int }_{1/2}^2|{\log }_{10}x|dx=$

• A. ${\log }_{10}\left(\frac{8}{e}\right)$
• B. $\frac{1}{2}{\log }_{10}\left(\frac{8}{e}\right)$
• C. $lo{g}_{10}\left(\frac{2}{e}\right)$
• D. none of these.

Answer 1

Answer: (B)

$\int logxdx=xlogx-x$
[By Using integration by parts]
$logxis-ve.forx,<1$ and $+ve$ for $x>1$.
Also $lo{g}_{10}x=\frac{logx}{log10}$
$\therefore$ given integral
$=\frac{1}{log10}\left[\underset{1/2}{\overset{1}{\int }}-logxdx+\underset{1}{\overset{2}{\int }}logxdx\right]$
$=-\frac{1}{log10}{\left[xlogx-x\right]}_{1/2}^1+\frac{1}{log10}{\left[xlogx-x\right]}_1^2$
$=-\frac{1}{log10}\left[-1-\frac{1}{2}log\frac{1}{2}+\frac{1}{2}\right]+\frac{1}{log10}\left[2log2-2+1\right]$
$=\frac{1}{log10}\left[\frac{1}{2}-\frac{1}{2}log2\right]+\frac{2log2-1}{log10}$
$=\frac{4log2-2+1-log2}{2log10}=\frac{3log2-1}{2log10}$
$=\frac{1}{2}\frac{log8-loge}{log10}=\frac{1}{2}\frac{log\left(\frac{8}{e}\right)}{log10}$
$=\frac{1}{2}lo{g}_{10}\left(\frac{8}{e}\right)$