TITLE: Find the sign of $\int_{0}^{2 \pi}\frac{\sin x}{x} dx$
QUESTION [8 upvotes]: I'd love your help with finding the sign of the following integral: $$\int_{0}^{2 \pi}\frac{\sin x}{x} dx$$
I know that computing it is impossible.  I tried to use integration by parts and maybe to learn about the sign of each part and conclude something but It didn't work for me.
Any suggestions?

REPLY [29 votes]: \begin{align*}
\int_0^{2\pi}\frac{\sin x}{x}\,dx&=\int_0^{\pi}\frac{\sin x}{x}\,dx+\int_\pi^{2\pi}\frac{\sin x}{x}\,dx\\
&=\int_0^{\pi}\frac{\sin x}{x}\,dx+\int_0^{\pi}\frac{\sin(x+\pi)}{x+\pi}\,dx\\
&=\int_0^{\pi}\Bigl(\frac{1}{x}-\frac{1}{x+\pi}\Bigr)\sin x\,dx\\
&=\pi\int_0^{\pi}\frac{\sin x}{x(x+\pi)}\,dx\\
&>0
\end{align*}