[Math] Find the sign of $\int_{0}^{2 \pi}\frac{\sin x}{x} dx$

Question

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?

Answer 1

\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*}