laplace transformation of a function using definition

Question

I want to find the laplace transformation of $x^ne^{ax}$ using the definition. I'm stuck with the integral. How shall I proceed the integral and find the final answer with $n$?

Answer 1

Hint: use the fact that

$$\int_0^{\infty} dy \, y^m \, e^{-b y} = \frac{\Gamma(m+1)}{b^{m+1}} = \frac{m!}{b^{m+1}}$$

when $\operatorname{Re}{b} \gt 0$ and $m \gt -1$

If you need to derive this when $m \in \mathbb{Z}$, use integration by parts.

Answer 2

For $s > a$,

\begin{align}\mathcal{L}(x^n e^{ax})(s) &= \int_0^\infty x^n e^{ax} e^{-sx}\, dx \\ &= \int_0^\infty x^n e^{-(s- a)x}\, dx\\ &= \frac{1}{(s - a)^{n+1}} \int_0^\infty u^n e^{-u}\, du \quad [u = (s - a)x]\\ &= \frac{n!}{(s - a)^{n+1}}. \end{align}

For $s \le a$, the laplace transform does not exist. For if $s \le a$, $x^n e^{ax} e^{-sx} = x^n e^{(a - s)x} \ge x^n$ on $[0, \infty)$ and $\int_0^\infty x^n\, dx$ diverges.