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Let $A = \int_{2013}^{2014} \frac{\sin{x}}{x} dx$.

Which of the following statement is true?

1) $A > 0$

2) $A = 0$

3) $A < 0$

From this Mathematica plot I see that it has to be $1)$ but I am not quite sure how to prove it analitically.

Sin(x)/x from 2013 to 2014

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Well, what do you think? Is $\frac{\sin x}{x}$ mostly positive or mostly negative between $2013$ and $2014$? – Arthur Jan 20 '14 at 12:32
It's customary to show your own thoughts. – TZakrevskiy Jan 20 '14 at 12:32
up vote 1 down vote accepted

Split it into two, with $r$ being the zero at about $2013.75$, and note that $$ \int_{2013}^{r}\frac{\sin x}{x}dx > \int_{2013}^{r}\frac{\sin x}{2014}dx $$ and $$ \int_{r}^{2014}\frac{\sin x}{x}dx > \int_{r}^{2014}\frac{\sin x}{2013}dx $$ so the sum of the two right-hand-side integrals is smaller than your integral, and still the sum of them is positive.

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