# How do I calculate $\int_{-\infty}^{\infty} e^{-ax^2} \;\mathrm{d}x$ for $a>0$ [duplicate]

This question already has an answer here:

$$\int_{-\infty}^{\infty} e^{-ax^2} \;\mathrm{d}x \quad\text{for } a>0$$

I don't even know where to begin to be honest so I haven't made any progress on it. The answer is $\sqrt{\pi/a}$.

Is it possible to solve this without using polar coordinates nor multiple integrals?

## marked as duplicate by user147263, user223391, TravisJ, user99914, Jonas MeyerMay 24 '15 at 4:11

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• this is a famous integral with no anti-derivative representable in terms of special functions, you could probably just google it – Thoth May 23 '15 at 21:03
• See this. – David Mitra May 23 '15 at 21:07
• You probably know a standard result about $\int_{-\infty}^\infty e^{-x^2/2}\,dx$, or some related integral. The one mentioned above has value $\sqrt{2\pi}$. One can then evaluate your integral by making a change of variable. – André Nicolas May 23 '15 at 21:08
• You may find this helpful also. – David Mitra May 23 '15 at 21:10
• Is it possible to solve this without polar coordinates nor integrating over both x and y? It was on the list of calculus problems my teacher asked the class to do, but I haven't learned any of those things yet so there's either other way to solve it or he wrongly put it there. I'll edit the original post to include this question. – xsr May 23 '15 at 21:17

## 1 Answer

There are many different ways of computing this integral. This article presents $16$ different ways of computing this integral.