Prove that $\int_{1}^{\infty}{\frac{e^{-x}}{\sqrt{1+x^{2}}}\text{dx}}\leq \frac{\sqrt{\pi}}{2\sqrt{2}e}$

Question

using inequalities of $L^p$ space how can we prove the following inequality:

$$\int_{1}^{\infty}{\frac{e^{-x}}{\sqrt{1+x^{2}}}\text{dx}}\leq \frac{\sqrt{\pi}}{2\sqrt{2}e}$$