Calculate the integral by using Fubini's theorem

Question

Exercise Compute integral $$\int_0^{\infty} \frac{e^{-{(3x)}^2}- e^{-{(6x)}^2}}{x} dx$$

My attempt and questions:

So we can write it as $$\frac{e^{-{(tx)}^2}}{x} |_6^3 \Rightarrow \int_0^{\infty}dx \int_6^3 -2txe^{-(tx)^2}dt$$

We can change order of intigration By Funini's theorem if function is positive which clearly is not as we have minus in front or if we show that $$\int\int_{[0,{\infty}]x[6,3] }2txe^{-(tx)^2} = \int_0^{\infty}dx \int_6^3 |-2txe^{-(tx)^2}| dt = \int_0^{\infty}dx \int_6^3 2txe^{-(tx)^2} dt < \infty$$

$$2 \int_0^{\infty} x dx \int_6^3 te^{-(tx)^2} dt =^{per partes} ln{\frac{1}{2}} < \infty$$

My question Did i calculate corrrectly?

Solving further Hence by Fubini's theore, we can change order of intigration:

$$\int_6^3 -2t dt \int_0^{\infty} xe^{-(tx)^2}dx$$

And here i am stuck because i get infinity by solving by the help of pwr partes.

My questionCould somone please help to proceed?

Answer 1

Note that we have

$$\begin{align} \int_0^\infty \frac{e^{-9x^2}-e^{-36x^2}}{x}\,dx&=2\int_0^\infty\int_3^6 ye^{-xy^2} \,dy\,dx\\\\ &=2\int_3^6\int_0^\infty ye^{-xy^2} \,dx\,dy\\\\ &=2 \int_3^6 \frac1y\,dy \\\\ &=2\log(2) \end{align}$$