Question

The value of ${\int }_a^b\frac{\log x}{x}dx$ is

• A. $\log \left(ab\right)\log \left(\frac{b}{a}\right)$
• B. $\frac{1}{2}\log \left(ab\right)\log \left(\frac{b}{a}\right)$
• C. $\log (a^2-b^2)$
• D. $(a+b)\log (a+b)$

Answer 1

Answer: (B)

$\frac{1}{2}\log \left(ab\right)\log \left(\frac{b}{a}\right)$

$I={\int }_a^b\frac{\log x}{x}dx={[\log x.\log x]}_a^b-{\int }_a^b\frac{\log x}{x}dx\Rightarrow 2I={\left[{\left(\log x\right)}^2\right]}_a^b={\left(\log b\right)}^2-{\left(\log a\right)}^2$
$\Rightarrow I=\frac{1}{2}(\log b+\log a)(\log b-\log a)=\frac{1}{2}\log ab\log \frac{b}{a}$