Question

The area bounded by the curve $y=f\left(x\right)$ the ordinates $x=1$ and $x=e^a(a>0)$ and the $x-$axis is given by $a{e}^a$. Then $f\left(x\right)$ is equal to $x+x\ln \left|x\right|$.

• A. True
• B. False

Answer 1

The correct option is B False

$\because {\int }_1^{e^a}f\left(x\right)dx=a{e}^a$
Differentiating both sides w.r.t $a$,we get
$f\left(e^a\right)\times e^a\ln e=a{e}^a\ln e+e^a.1=(a+1)\ln e^a$
$\Rightarrow f\left(e^a\right)=\frac{(a+1)\ln e^a}{\ln e^a}$
$\therefore f\left(e^a\right)=a+1$
Replacing $a$ by $\ln x$, then $f\left(x\right)=\ln \left|x\right|+1$