Question

What is ${\int }_1^2\ln xdx$ equal to?

• A. $\ln 2$
• B. $1$
• C. $\ln \left(\frac{4}{e}\right)$
• D. $\ln \left(\frac{e}{4}\right)$

Answer 1

The correct option is C $\ln \left(\frac{4}{e}\right)$

Let, $y=\ln x$ then, $x=e^y$

Now at, $x=1,y=0$

$x=2,y=\ln 2$

$\Rightarrow dx=e^y.dy$

Putting the values, we get

${\int }_1^2\ln xdx={\int }_0^{\ln 2}{e}^y.y.dy$

$={[y.e^y]}_0^{\ln 2}-{\int }_0^{\ln 2}{e}^{y}dy$

$=2\ln 2-1$

$=2\ln 2-\ln \left(e\right)$

$=\ln \left(\frac{4}{e}\right)$

Hence, C is correct.