I need to prove that the following inequality holds:
$\int_{0}^{e} \sqrt{e^x-1} + \int_{0}^{e} \log{(x^2+1)} \geq e^2$
Any support is welcome. Thanks.
I need to prove that the following inequality holds:
$\int_{0}^{e} \sqrt{e^x-1} + \int_{0}^{e} \log{(x^2+1)} \geq e^2$
Any support is welcome. Thanks.
Note that the inverse function of $f(x)=\sqrt{e^x-1}$, $0\le x\le e$, is $f^{-1}(x)=\log(x^2+1)$. So, you may apply the appropriate version of Young's Inequality.