Prove: $ \Big|\int_1^\sqrt{3} \! \frac{\sin(x)}{e^x(x^2 +1)} \, \mathrm{d} x \Big| \leq \frac{\pi}{12e}$
Approach: we know:
$ \Big|\int_1^\sqrt{3} \! \frac{\sin(x)}{e^x(x^2 +1)} \, \mathrm{d} x \Big| \leq \int_1^\sqrt{3} \! \frac{|\sin(x)|}{e^x(x^2 +1)} \, \mathrm{d} x \leq \int_1^\sqrt{3} \! \frac{1}{e^x(x^2 +1)} \, \mathrm{d} x$ Since $|\sin(x)|$ is bounded by 1 for every $x$. Can we keep estimating this integral, or should we just solve the last integral?
thanks.