Step 1

Given Laplace transform is \(F(s)=\frac{4s+5}{s^{2}+9}

The function can be obtained as follows. \(f(t)=L^{-1}\left\{F(s)\right\}

\(=L^{-1}\left\{\frac{4s+5}{s^{2}+9}\right\}

\(=L^{-1}\left\{\frac{4s}{s^2+9}\right\}+L^{-1}\left\{\frac{5}{s^2+9}\right\}

Step 2 On further simplifications, \(f(t)=4L^{-1}\left\{\frac{s}{s^2+3^{2}}\right\}+\frac{5}{3}L^{-1}\left\{\frac{3}{s^2+3^{2}}\right\}

\(=4\cos(3t)+\frac{5}{3}\sin(3t)\)

Given Laplace transform is \(F(s)=\frac{4s+5}{s^{2}+9}

The function can be obtained as follows. \(f(t)=L^{-1}\left\{F(s)\right\}

\(=L^{-1}\left\{\frac{4s+5}{s^{2}+9}\right\}

\(=L^{-1}\left\{\frac{4s}{s^2+9}\right\}+L^{-1}\left\{\frac{5}{s^2+9}\right\}

Step 2 On further simplifications, \(f(t)=4L^{-1}\left\{\frac{s}{s^2+3^{2}}\right\}+\frac{5}{3}L^{-1}\left\{\frac{3}{s^2+3^{2}}\right\}

\(=4\cos(3t)+\frac{5}{3}\sin(3t)\)