TITLE: How prove this $\int_{0}^{2\pi}e^{\sin{x}}dx<2\pi e^{\frac{1}{4}}$
QUESTION [6 upvotes]: Show that this intergral inequality $$\int_{0}^{2\pi}e^{\sin{x}}dx<2\pi e^{\frac{1}{4}}$$
I know this use Taylor's formula.But I think is very ugly,maybe this problem have simple methods.Thank you

REPLY [6 votes]: Use that $\cosh(x) \leq (\cosh(1)-1) x^2 + 1$ for all $x\in[-1,1]$ to obtain the sharper estimate $$ \int_0^{2\pi}e^{\sin(x)}dx=2 \int_{-\pi/2}^{\pi/2}\cosh(\sin(x))dx\leq \pi\left(1+\cosh(1)\right).$$ Now the question remains to give a simple proof of the inequality $$1+\cosh(1) < 2e^{1/4}.$$