Question

If $\underset{1}{\overset{y}{\int }}x\ln xdx=\frac{1}{4}$, then the value of $y$ is equal to

• A. e
• B. $\sqrt{e}$
• C. $\frac{1}{\sqrt{e}}$
• D. $\frac{1}{e}$

Answer 1

Answer: (B)

Applying integration by parts with $u=\ln x$ and $v=\frac{x^2}{2}$, we get
$\underset{1}{\overset{y}{\int }}x\ln xdx={\frac{x^2}{2}\ln x|}_1^y-\frac{1}{2}\underset{1}{\overset{y}{\int }}xdx=\frac{y^2\ln y}{2}-\frac{y^2}{4}+\frac{1}{4}.$
So, $y^2\ln y=\frac{y^2}{2}$. Since $y>1$, we obtain $\ln y=\frac{1}{2}$ and thus $y=\sqrt{e}$.
Note: $y=0$ is also acceptable