Question

Evaluate the integrals ${\int }_a^b\frac{\log x}{x}dx$

• A. $\frac{1}{2}\log \left(ab\right)\cdot \log \frac{b}{a}$
• B. $\frac{1}{4}\log \left(ab\right)\cdot \log \frac{b}{a}$
• C. $\log \left(ab\right)\cdot \log \frac{b}{a}$
• D. $-\frac{1}{2}\log \left(ab\right)\cdot \log \frac{b}{a}$

Answer 1

The correct option is A $\frac{1}{2}\log \left(ab\right)\cdot \log \frac{b}{a}$

Substitute $\log x=t\therefore \left(1/x\right)dx=dt$ and adjust the limits
$\therefore I=\int tdt={\left[\frac{1}{2}{t}^2\right]}_{\log a}^{\log b}$
$\therefore I=\frac{1}{2}[{\left(\log b\right)}^2-{\left(\log a\right)}^2]$
$=\frac{1}{2}\left[(\log b+\log a)(\log b-\log a)\right]$
$\therefore I=\frac{1}{2}\log \left(ab\right)\cdot \log \frac{b}{a}.$