1. Apply the Laplace transform

By the definition of Laplace transform, we have

\displaystyle\mathscr{L}\{t^n\}=\int_{t=0}^{\infty}t^ne^{-st}dt

2. Use integration by parts

Using integration by parts, we get:

\displaystyle\mathscr{L}\{t^n\}=\left[t^n\left(\frac{e^{-st}}{-s}\right)\right]_{t=0}^{\infty}-\int_{t=0}^{\infty}\left(\frac{e^{-st}}{-s}\right)\frac{d t^n}{dt}dt

Simplifying the derivative and taking same constants common

\displaystyle\mathscr{L}\{t^n\}=\frac{-1}{s}\left[t^n e^{-st}\right]_{t=0}^{\infty}+\frac{1}{s}\int_{t=0}^{\infty}e^{-st}n t^{n-1}dt

3. Simplify the obtained results

As we have stated above, that s is positive and large enough. So, the product t^n e^{-st} converge to zero as t\to\infty.

\displaystyle\mathscr{L}\{t^n\}=0+\frac{n}{s}\int_{t=0}^{\infty}e^{-st}t^{n-1}dt

In the above equation \displaystyle\int_{t=0}^{\infty}e^{-st}t^{n-1}dt is same as \displaystyle\int_{t=0}^{\infty}e^{-st}t^{n}dt exept n is replaced by n-1.

If

\displaystyle I_n=\int_{t=0}^{\infty}e^{-st}t^{n}dt,

then

\displaystyle I_{n-1}=\int_{t=0}^{\infty}e^{-st}t^{n-1}dt.

4. Most important part

The previous equation becomes:

\displaystyle\mathscr{L}\{t^n\}=I_n=\frac{n}{s}I_{n-1}

reducing the I_{n-1} by one

\displaystyle I_{n-1}=\frac{n-1}{s}I_{n-2}

similarly

\displaystyle I_{n-2}=\frac{n-2}{s}I_{n-3}

Similarly, we can write

\begin{aligned} \mathscr{L}\{t^n\}=&\frac{n}{s}I_{n-1}\\ =&\frac{n}{s}\frac{n-1}{s}I_{n-2}\\ =&\frac{n}{s}\frac{n-1}{s}\frac{n-2}{s}I_{n-3} \end{aligned}

5. Repeat the process

Do the same process for n times we get:

\displaystyle\mathscr{L}\{t^n\}=\frac{n}{s}\frac{n-1}{s}\frac{n-2}{s}\dots\frac{n-(n-1)}{s} I_{0}

these are n factors and

\displaystyle I_{0}=\int_{t=0}^{\infty}e^{-st}t^{0}dt=\mathscr{L}\{1\}

yields:

\displaystyle\mathscr{L}\{t^n\}=\frac{n}{s}\frac{n-1}{s}\frac{n-2}{s}\dots\frac{1}{s} \mathscr{L}\{1\}

Multiplying the factors, and putting the value of \mathscr{L}\{1\}, we get:

\displaystyle \mathscr{L}\{t^n\}=\frac{n!}{s^n} \frac{1}{s}

6. Result

simplifying the above expression, the Laplace transform of the function f(t)=t^n is given by:

\displaystyle\mathcal{L}\{t^n\}=\frac{n!}{s^{n+1}}

We reached the end of this exercise about calculating the Laplace transform of t to the power of n using the definition.