Question

USing f(x)=sinx/x
prove that
\[ \large \int\limits^{\pi/2}_{0}e^{-\sin x}dx<(\pi/2R)(1-e^{-R}) for R>0\]

Answer 1

@ash2326 , @badreferences

Answer 2

Alright, squeeze theorem.\[\int_0^{\frac\pi2}e^{-\sin x}\,dx\]Since the value for \(e^{-\sin x}\) fluctuates between \(e^x\) and \(\frac1{e^x}\), the area underneath the curve fluctuates as well. Can you guess where I want to go from here?

Answer 3

"Fluctuates" is the wrong word. I meant to say "high" and "low" bounds. Sorry/