A double integral.

Question

There's a question in my brother's textbook which asks to evaluate the following integral by change of order of integration :
$$ \int \limits_{0}^\infty \int \limits_{0}^x xe^{-\frac{x^3}{y}} \,dy \,dx $$ So, the problem here is that as I proceeded after the change of order ,I used the incomplete gamma function but my brother doesn't know about them neither does his book mentions about it.
So the question is how to solve without using the incomplete gamma function.

Answer 1

Too long for a comment

Changing the order of integration $$I=\int_0^\infty dy\int_y^\infty e^{-\frac{x^3}{y}} xdx\stackrel{x=yt}{=}\int_0^\infty y^2dy\int_1^\infty e^{-y^2t^3}tdt$$ Changing the order once again and making the substitution $y^2=x$ $$I=\frac{1}{2}\int_1^\infty tdt\int_0^\infty \sqrt xe^{-xt^3}dx$$ Making another substitution $y=xt^3$ $$I=\frac{1}{2}\int_1^\infty tdt\int_0^\infty t^{-\frac{9}{2}}\sqrt ye^{-y}dy=\frac{\Gamma\Big(\frac{3}{2}\Big)}{2}\int_1^\infty t^{-\frac{7}{2}}dt=\frac{\Gamma\Big(\frac{3}{2}\Big)}{2}\frac{2}{5}=\frac{\sqrt\pi}{10}$$