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Prove that $\displaystyle\left| \int_1^{\sqrt{3}}\frac{e^{-x}\sin x}{x^2+1}dx\right|\le \frac{\pi}{12e}$

I tried comparing this integral with $\displaystyle\int \frac{e^{-x}}{x^2} dx$ but this integrates out to something greater than $\frac{\pi}{12e}$. Any hints would be appreciated, as this is an exercise I'm working on to review calculus. Thank you!

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up vote 3 down vote accepted

*(Very big) hint:*$$ \int_1^\sqrt{3} \frac{dx}{1+x^2} = \dfrac{\pi}{12} $$

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