3

Find the integral $$\int_{0}^{\infty}\frac{ e^{-ax^{2}}-e^{-bx^{2}} }{x}dx,$$ $a>0,\ b>0$.

I tried to split the integral and use Feynman substitution, but I got an answer that doesn't look like the right one.

I considered this integral $$\int_{0}^{∞}dy\int_{0}^{∞}e^{-ax^{2}-xy}dx.$$

Gary
  • 31,845
Maksim S
  • 31

2 Answers2

3

This is a Frullani integral with the choice $f(x) = e^{-x^2}$, but you have to be careful about choosing $a$ and $b$, because the "$a$" and "$b$" in your question are not the same as those in the Wikipedia article.

heropup
  • 135,869
  • $\int_{0}^{\infty}\frac{ e^{-ax^{n}}-e^{-bx^{n}} }{x}dx=\frac{1}{n}\int_{0}^{\infty}\frac{ e^{-at}-e^{-bt} }{t}dt$ – Svyatoslav Feb 11 '22 at 12:33
3

Let $J(t)=\int_{0}^{\infty}\frac{ e^{-tx ^{2}}-e^{-x^2}}{x}dx$. Then $J’(t)= -\int_{0}^{\infty}x e^{-tx ^{2}}dx=-\frac1{2t} $ and $$\int_{0}^{\infty}\frac{ e^{-ax^{2}}-e^{-bx^{2}} }{x}dx =J(a)-J(b)=\int_b^a J’(t)dt= \frac12\ln \frac ba $$

Quanto
  • 97,352