# Find the laplace transform of the following: a) t^2 sin kt b) tsin kt

Find the laplace transform of the following:
$a\right){t}^{2}\mathrm{sin}kt$
$b\right)t\mathrm{sin}kt$
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$\text{Step 1}$
$\text{Recall the following.}$
$L\left\{{t}^{k}f\left(t\right)\right\}=\left(-1{\right)}^{k}\frac{{d}^{k}}{d{s}^{k}}\left(L\left\{f\left(t\right)\right\}\right)$
$L\left\{\mathrm{sin}\left(at\right)\right\}=\frac{a}{{s}^{2}+{a}^{2}}$
$\text{Step 2}$
$\text{(a) Obtain the Laplace transform as follows.}$
$\left(a\right)f\left(t\right)={t}^{2}\mathrm{sin}\left(kt\right)$
$L\left\{\mathrm{sin}\left(kt\right)\right\}=\frac{k}{{s}^{2}+{k}^{2}}$
$L\left\{{t}^{2}\mathrm{sin}\left(kt\right)\right\}=\left(-1{\right)}^{2}\frac{{d}^{2}}{d{s}^{2}}\left(L\left\{\mathrm{sin}\left(kt\right)\right\}\right)$
$=\frac{{d}^{2}}{d{s}^{2}}\left(\frac{k}{{s}^{2}+{k}^{2}}\right)$
$=\frac{d}{ds}\left(\frac{d}{ds}\left(\frac{k}{{s}^{2}+{k}^{2}}\right)\right)$
$=\frac{d}{ds}\left(-\frac{2ks}{\left({s}^{2}+{k}^{2}{\right)}^{2}}\right)$
$=-2k\frac{d}{ds}\left(\frac{s}{\left({s}^{2}+{k}^{2}{\right)}^{2}}\right)$
$=-2k\frac{\left({s}^{2}+{k}^{2}{\right)}^{2}\frac{d}{ds}\left(s\right)-s\frac{d}{ds}\left[\left({s}^{2}+{k}^{2}{\right)}^{2}\right]}{\left(\left({s}^{2}+{k}^{2}{\right)}^{2}{\right)}^{2}}$
$=-2k\frac{\left({s}^{2}+{k}^{2}{\right)}^{2}-4{s}^{2}\left[\left({s}^{2}+{k}^{2}\right)\right]}{\left(\left({s}^{2}+{k}^{2}{\right)}^{2}{\right)}^{2}}$
$=-2k\frac{\left({s}^{2}+{k}^{2}\right)-4{s}^{2}}{\left({s}^{2}+{k}^{2}{\right)}^{3}}$
$=\frac{2k\left(3{s}^{2}-{k}^{2}\right)}{\left({s}^{2}+{k}^{2}{\right)}^{3}}$
$\text{Step 3}$
$\text{(b) Obtain the Laplace transform as follows.}$
$\left(b\right)f\left(t\right)=t\mathrm{sin}\left(kt\right)$
$L\mathrm{sin}\left(kt\right)=\frac{k}{{s}^{2}+{k}^{2}}$
$L\left\{t\mathrm{sin}\left(kt\right)\right\}=\left(-1{\right)}^{1}\frac{{d}^{1}}{d{s}^{1}}\left(L\left\{\mathrm{sin}\left(kt\right)\right\}\right)$
$=-\frac{d}{ds}\left(\frac{k}{{s}^{2}+{k}^{2}}\right)$
$=-k\frac{d}{ds}\left(\frac{1}{{s}^{2}+{k}^{2}}\right)$
$=-k\left(\frac{-1}{\left({s}^{2}+{k}^{2}{\right)}^{2}}\right)\frac{d}{ds}\left({s}^{2}+{k}^{2}\right)$

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