# Please provide steps The inverse Laplace transform for displaystyle{F}{left({s}right)}=frac{8}{{{s}+{9}}}-frac{6}{{{s}^{2}-sqrt{{3}}}} is a) displayst

Laplace transform
The inverse Laplace transform for
$$\displaystyle{F}{\left({s}\right)}=\frac{8}{{{s}+{9}}}-\frac{6}{{{s}^{2}-\sqrt{{3}}}}$$ is
a) $$\displaystyle{8}{e}^{{-{9}{t}}}-{6} \sin{{h}}{{\left({3}{t}\right)}}$$
b) $$\displaystyle{8}{e}^{{-{9}{t}}}-{6} \cos{{h}}{\left({3}{t}\right)}$$
c) $$\displaystyle{8}{e}^{{{9}{t}}}-{6} \sin{{h}}{\left({3}{t}\right)}$$
d) $$\displaystyle{8}{e}^{{{9}{t}}}-{6} \cos{{h}}{\left({3}{t}\right)}$$

2020-10-26
Step 1
Consider the function:
$$\displaystyle{F}{\left({s}\right)}=\frac{8}{{{x}+{9}}}-\frac{6}{{{x}^{2}-\sqrt{{3}}}}$$
Find Inverse Laplace Transform.
Apply Partial fraction on $$\displaystyle\frac{6}{{{x}^{2}-\sqrt{{3}}}}$$
$$\displaystyle\frac{6}{{{x}^{2}-\sqrt{{3}}}}=\frac{A}{{{x}-{\sqrt[{{4}}]{{{3}}}}}}+\frac{B}{{{x}+{\sqrt[{{4}}]{{{3}}}}}}$$
$$\displaystyle{6}={A}{\left({x}+{\sqrt[{{4}}]{{{3}}}}\right)}+{B}{\left({x}-{\sqrt[{{4}}]{{{3}}}}\right)}$$
Let $$\displaystyle{x}={\sqrt[{{4}}]{{{3}}}}\ \text{ and }\ {x}=-{\sqrt[{{4}}]{{{3}}}}$$ and find A and B
$$\displaystyle{A}=\frac{3}{{{\sqrt[{{4}}]{{{3}}}}}}$$
$$\displaystyle{B}=-\frac{3}{{{\sqrt[{{4}}]{{{3}}}}}}$$
$$\displaystyle{F}{\left({s}\right)}=\frac{8}{{{x}+{9}}}-\frac{3}{{{\sqrt[{{4}}]{{{3}}}}{\left({x}-{\sqrt[{{4}}]{{{3}}}}\right)}}}+\frac{3}{{{\sqrt[{{4}}]{{{3}}}}{\left({x}+{\sqrt[{{4}}]{{{3}}}}\right)}}}$$
Step 2
Inverse Laplace transform property
$$\displaystyle{L}^{ -{{1}}}{\left\lbrace{a} f{{\left({s}\right)}}+ g{{\left({s}\right)}}\right\rbrace}={a}{L}^{ -{{1}}}{\left\lbrace f{{\left({s}\right)}}\right\rbrace}+{L}^{ -{{1}}}{\left\lbrace g{{\left({s}\right)}}\right\rbrace}\ \text{ and }\ {L}^{ -{{1}}}{1}{s}+{a}={e}^{{-{a}{t}}}$$
$$\displaystyle{L}^{ -{{1}}}{\left\lbrace\frac{8}{{{x}+{9}}}-\frac{3}{{{\sqrt[{{4}}]{{{3}}}}{\left({x}-{\sqrt[{{4}}]{{{3}}}}\right)}}}+\frac{3}{{{\sqrt[{{4}}]{{{3}}}}{\left({x}+{\sqrt[{{4}}]{{{3}}}}\right)}}}\right\rbrace}={8}{L}^{ -{{1}}}{\left\lbrace\frac{1}{{{x}+{9}}}\right\rbrace}-{3}^{{\frac{3}{{4}}}}{L}^{ -{{1}}}{\left\lbrace\frac{1}{{{x}-{\sqrt[{{4}}]{{{3}}}}}}\right\rbrace}+{3}^{{\frac{3}{{4}}}}{L}^{ -{{1}}}{\left\lbrace\frac{1}{{{x}+{\sqrt[{{4}}]{{{3}}}}}}\right\rbrace}$$
$$\displaystyle={8}{e}^{{-{9}{t}}}-{3}^{{\frac{3}{{4}}}}{e}^{{{\sqrt[{{4}}]{{{3}}}}{t}}}+{3}^{{\frac{3}{{4}}}}{e}^{{-{\sqrt[{{4}}]{{{3}}}}{t}}}$$