Upper bound $\int_0^\infty \exp\left(-\alpha(x(x+\beta+1))^2\right)$

Question

I am looking for an upper bound as tight as possible going to $0$ as $\alpha\to\infty$ of

$$\int_0^\infty \exp\left(-\alpha(x(x+\beta+1))^2\right).$$

Answer 1

Hint

Let $\beta >0$. Remark that $$\int_0^\infty e^{-a\big(x(x+\beta +1)\big)^2}\,\mathrm d x\leq \int_0^\infty e^{-a(\beta +1)^2x^2}\,\mathrm d x,$$ and use the fact that $$\int_0^\infty e^{-x^2}\,\mathrm d x=\frac{\sqrt\pi}{2}.$$