im trying to derive laplace of $\mathcal{L}\left\{ t^n \right\}$
sorry my mathjax skills are bad i tried my best to write it clear.
i can solve the equation by following with khan academy video however i wanted to solve it
with my own way and i find trouble in doing that didnt practice math in long time so im restudying calculus again:
$$ \mathcal{L}\left\{ t^n \right\} = \int\limits_{0}^{\infty}t^{n}e^{-st}\;\mathrm{d}t $$
$$ let \text{ } {u} = e^{-st}$$
$$ du = -s e^{-st} dt$$
$$ let \text{ } {dv} = t^{n}dt$$
$$ v = \frac{t^{n+1}}{n+1}$$
intgration by parts $$ \int{uv'} = uv - \int{vu'}$$
$$ [e^{-st} * \frac{t^{n+1}}{n+1}] \Big|_0^\infty - \int\limits_0^\infty {\frac{t^{n+1}}{n+1}*(-s*e^{-st})dt}$$
$$ 0 + \frac{s}{n+1} \int_0^\infty{t^{n+1} * e^{-st}\text{ }\mathrm{d}t}$$
$$ \frac{s}{n+1} \mathcal{L}{(t^{n+1})}$$
$$ \frac{s}{n+1} * \frac{s}{n+2} * \mathcal{L}{(t^{n+2})}$$
$$ \frac{s}{n+1} * \frac{s}{n+2} * \frac{s}{n+3} * \mathcal{L}{(t^{n+3})}$$
where is my mistake ?
well, actually you are very close to a solution. let's notice that $$ \mathcal{L}\{1\}(s)=\intop_0^{\infty}e^{-st}dt=-\frac{1}{s}\left[e^{-st}\right]_{t=0}^{t=\infty}=\frac{1}{s} $$ you have proven already that $\mathcal{L}\{t^n\}=\frac{s}{n+1}\cdot\mathcal{L}\{t^{n+1}\}$, so we can guess that the general formula will be $\mathcal{L}\{t^n\}=\frac{n!}{s^{n+1}}$ and prove it by induction.
the base case $n=0$: we've already proven
induction step $n\Rightarrow n+1$: let us assume that $\mathcal{L}\{t^n\}=\frac{n!}{s^{n+1}}$ and prove that $\mathcal{L}\{t^{n+1}\}=\frac{(n+1)!}{s^{n+2}}$.
as you proved, we know that $\mathcal{L}\{t^n\}=\frac{s}{n+1}\cdot\mathcal{L}\{t^{n+1}\}$, so by our induction hypothesis we can conclude that $\frac{s}{n+1}\cdot\mathcal{L}\{t^{n+1}\}=\frac{n!}{s^{n+1}}$ and after dividing by $\frac{s}{n+1}$ we will get $\mathcal{L}\{t^{n+1}\}=\frac{n+1}{s}\cdot\frac{n!}{s^{n+1}}=\frac{(n+1)!}{s^{n+2}}$ as desired.