Question

The value of the integral ${\int }_2^4$ $\left(\frac{\log t}{t}\right)dt$ is equal to

• A. $\frac{1}{2}{\left(\log 2\right)}^2$
• B. $\frac{5}{2}{\left(\log 2\right)}^2$
• C. $\frac{3}{2}{\left(\log 2\right)}^2$
• D. ${\left(\log 2\right)}^2$
• E. $\frac{3}{2}\left(\log 2\right)$

Answer 1

Answer: (C)

Let $I={\int }_2^4\left(\frac{\log t}{t}\right)dt$
Let $\log t=z$
$\Rightarrow \frac{1}{t}dt=dz$
$\therefore I={\int }_{\log 2}^{\log 4}Zdz={\left[\frac{z^2}{2}\right]}_{\log 2}^{\log 4}$
$=\frac{1}{2}\left[(\log 4{)}^2-(\log 2{)}^2\right]$
$=\frac{1}{2}\left[(2\log 2{)}^2-(\log 2{)}^2\right]$
$=\frac{3}{2}(\log 2{)}^2$