Finding which one is a larger expression

Question

How to find which expression is larger: $$e^{\pi\cdot \int_0^1\sqrt {\tan(x)}dx}$$ $$or$$ $$ \pi^{e\cdot \int_0^1\sin^2(x)dx}$$

This question seems bit complicated. What is the best way to start off analytically without using a calculator?

Answer 1

Since on the interval $(0,1)$ we have $\sin(x)\leq x\leq\tan(x)$ and $2<e<3<\pi<4$,

$$ e^{\pi\int_{0}^{1}\sqrt{\tan x}\,dx} \geq e^{\pi \int_{0}^{1}\sqrt{x}\,dx} = e^{2\pi/3} \geq e^2 >4 $$ while $$ \pi^{e\int_{0}^{1}\sin(x)^2\,dx} \leq \pi^{e/3} \leq \pi < 4.$$

The comparison is not difficult, since the difference between such numbers is pretty large.