Question

The value of the integral $\displaystyle \int_2^4$ $\left(\frac{\log\, t }{t}\right) dt$ 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 $I=\displaystyle\int_{2}^{4}\left(\frac{\log t}{t}\right) d t$
Let $\log\, t=z $
$\Rightarrow \frac{1}{t} d t=d z$
$\therefore I=\displaystyle\int_{\log 2}^{\log 4} Z\, d z=\left[\frac{z^{2}}{2}\right]_{\log 2}^{\log 4}$
$=\frac{1}{2}\left[(\log 4)^{2}-(\log 2)^{2}\right]$
$=\frac{1}{2}\left[(2 \log 2)^{2}-(\log 2)^{2}\right]$
$=\frac{3}{2}(\log 2)^{2}$