Find $\int_{-\infty}^{\infty} e^{-\lambda(x - a)^2} dx$

Question

I've tried integrating by parts and using polar coordinates but I couldn't solve it. Heck, I've even tried a simpler integral of e^(-x² + x) from -∞ to +∞ in the Wolfram Alpha app but it doesn't tell me how it solved it.

Answer 1

Let $u=\sqrt{\lambda}(x-a)$,

$$\int_{-\infty}^{\infty} e^{-\lambda(x - a)^2} dx =\frac1{\sqrt{\lambda}}\int_{-\infty}^{\infty} e^{-u^2} du = \frac{\sqrt\pi}{\sqrt{\lambda}}$$

where,

$$\left(\int_{-\infty}^{\infty} e^{-u^2} du\right)^2=\int_0^{2\pi}\int_0^\infty e^{-r^2}rdrd\theta=\pi$$

Answer 2

Hint: Substituting $$t=x-a$$ then we have $$dt=dx$$ and our integral is given by $$\int e^{-\lambda t^2}dt=\frac{\sqrt{\pi } \text{erf}\left(\sqrt{\lambda } t\right)}{2 \sqrt{\lambda }}$$