Question

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

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

Answer 1

The correct option is C

$\frac{3}{2}\log {\left(2\right)}^2$

Explanation for the correct option:

Compute the required value.

Given: ${\int }_2^4\frac{\log \left(t\right)}{t}dt$

Let,

$$\begin{gathered} \log \left(t\right)=x \\ \frac{1}{t}dt=dx \end{gathered}$$

When $t=2,x=2$

$\Rightarrow t=\log \left(4\right),x=\log \left(2\right)$

$$\begin{gathered} {\int }_{\log \left(2\right)}^{\log \left(4\right)}xdx={\left|\frac{x^2}{2}\right|}_{\log \left(2\right)}^{\log \left(4\right)} \\ {\int }_{\log \left(2\right)}^{\log \left(4\right)}xdx=\frac{1}{2}\left|{\log }^2\left(4\right)-{\log }^2\left(2\right)\right| \\ {\int }_{\log \left(2\right)}^{\log \left(4\right)}xdx=\frac{1}{2}\left|\left(\log \left(4\right)-\log \left(2\right)\right)\left(\log \right(4)+\log (2\left)\right)\right| \\ {\int }_{\log \left(2\right)}^{\log \left(4\right)}xdx=\frac{1}{2}\left|\log (4\times 2)\log \left(\frac{4}{2}\right)\right|\left[\log \left(a\right)-\log \left(b\right)=\log \left(\frac{a}{b}\right),\log \left(m\right)+\log \left(n\right)=\log \left(mn\right)\right] \\ {\int }_{\log \left(2\right)}^{\log \left(4\right)}xdx=\frac{1}{2}\left|\log \left(8\right)\log \left(2\right)\right| \\ {\int }_{\log \left(2\right)}^{\log \left(4\right)}xdx=\frac{1}{2}\left|3\log \left(2\right)\log \left(2\right)\right| \\ {\int }_{\log \left(2\right)}^{\log \left(4\right)}xdx=\frac{3}{2}\left|{\log }^2\left(2\right)\right| \end{gathered}$$

Hence, option $C$ is the correct answer.