Question

${\int }_a^b\frac{\log x}{x}dx=.......$ (where $a,b\in R^{+}$)

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

Answer 1

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

$I={\int }_a^b\frac{\log x}{x}dx(a,b+R^{+})={\int }_a^b\log x\left(\frac{d}{dx}\log x\right)dx$
$=\frac{1}{2}{\left[{\left(\log x\right)}^2\right]}_a^b(\therefore \int {\left(f\right(x\left)\right)}^n{f}^{\prime }\left(x\right)dx=\frac{{\left(f\right(x\left)\right)}^{n+1}}{n+1})$
$=\frac{1}{2}[{\left(\log a\right)}^2-{\left(\log b\right)}^2]=\frac{1}{2}\left[(\log b+\log a)(\log b-\log a)\right]$
$\therefore I=\frac{1}{2}\log \left(ab\right)\log \left(\frac{b}{a}\right)$