# Laplace transform of simple function

I have the function below:

$$3-2t+3t^2$$

I tried to get the transfer function, but my solution seems to be wrong, can someone please tell me what I am doing wrong:

$$\frac{1}{s}+\frac{1}{s^3}$$

-

3
Are we just finding $\int_0^\infty (3-2t+3t^2)e^{-st}\,dt$? If so (using integration by parts for the last two terms) I think we get $\frac{3}{s}-\frac{2}{s^2}+\frac{6}{s^3}$. As to what you did wrong, that is hard to know if one does not see what you did. – André Nicolas May 31 '12 at 19:33
You can use the following properties: $\mathcal{L} \{1\} =\frac{1}{s}$ , $\mathcal{L} \{t^n \} =\frac{n!}{s^{n+1}}$ – passenger May 31 '12 at 19:37
It would be better if you added what was done in a clear way. Indeed, Andre got what you are looking for. – Babak S. May 31 '12 at 19:40
Is the function of $t$ meant to be the impulse response? – copper.hat May 31 '12 at 19:44
Thanks guys, this was so simple I was ashamed to continue the question :( got it right now. – Sean87 May 31 '12 at 20:51

## 2 Answers

Just looking at a table and seeing how $$\mathcal L\{1\}=\frac{1}{s},\quad \mathcal L\{t\}=\frac{1}{s^2},\quad \mathcal L\{t^2\}=\frac{2}{s^3}$$... add them up, combine the constants, hope to get something like $$F(s)=\frac{3}{s}-\frac{2}{s^2}+\frac{6}{s^3}$$

-

$$\begin{align*}f(t)&=3-2t+3t^2\\ \\ F(s)&=\mathcal L\{3-2t+3t^2\}\\ \\ &=\mathcal L\{3\}-\mathcal L\{2t\}+\mathcal L\{3t^2\}\\ \\ &=3\mathcal L\{t^0\}-2\mathcal L\{t^1\}+3\mathcal L\{t^2\}\\\\&=\end{align*}$$

use $$\boxed{\begin{align*}\mathcal L\{t^n\}&=\frac{\Gamma(n+1)}{s^{n+1}},\qquad n>-1\\&=\frac{n!}{s^{n+1}},\qquad \qquad n\in\mathbb Z> 0\qquad\end{align*}}$$

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 5, "mathjax_display_tex": 6, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9565728306770325, "perplexity_flag": "middle"}