$\int_0^\infty f^k (t)\cdot t^j \,dt$

Question

Calcular el $$\int_0^\infty f^{(k)}(t)\cdot t^j \,dt$$ for $$0\le j\le k$$ function values of f $$(f\in C_0 ^{\infty}[0,\infty))$$

Alguna idea de cómo calcular esta integral?

Answer 1

$$I(j,k)=|t^jf^{(k-1)}(t)|_{0}^{\infty}-jI(j-1,k-1) $$ So, if $\lim_{t\to \infty}t^{i+1}f^{(i)}(t)$ exists, and for an example, if it is $0$, then, $$I(j,k)=-jI(j-1,k-1)=(-1)^{j-1}j!f^{(k-j-1)}(0)$$