Sinal de integração e diferenciação

The technique of differentiation under the integral sign, popularly known as Feynman's trick, is a way to impress your friends and colleagues in dire circumstances:

Often, integrals, like $\int_0^\infty x^2 e^{-x} dx$, live naturally not in isolation, but in families. When you learn about differential equations, you see that a function can arise in more than one way, e.g. as explicit formula $y = e^x$ vs. as differential equation $y' = y$. There lies the philosophy behind Feynman's method: an integral may have a useful alternative description, perhaps as a derivative of another family of integrals, or as a double integral (integral of a family of integrals).

Finding that alternative description (in terms of families) is more of an art than an exact science. Here, $\int_0^\infty x^2 e^{-x} dx$ fits in the family $$I(n,t) = \int_0^\infty x^n e^{-tx} dx,$$ as the member $I(2,1)$. The derivative of $I(n,t)$ with respect to $t$ is $$\frac{d}{dt}I(n,t) = -\int_0^\infty x^{n+1} e^{-tx} dx,$$ a.k.a. just $-I(n+1,t)$! Thus $I(2,t)$ is just the second derivative of $I(0,t)$!

On the other hand, when $n=0$, we easily evaluate the integral as long as $t>0$. So, $$I(2,t) = I''(0,t) = \frac{2}{t^3},$$ and we find the integral we started with, $$\int_0^\infty x^2 e^{-x} dx = I(2,1) = \boxed{2}.$$