Question

Find the \(L^{-1} (\frac{s+3}{4s^2+9})\).

(a) \(\frac{1}{4} cos⁡(\frac{3t}{2})+\frac{1}{2} cos⁡(\frac{3t}{2})\)

(b) \(\frac{1}{4} cos⁡(\frac{3t}{4})+\frac{1}{2} sin⁡(\frac{3t}{2})\)

(c) \(\frac{1}{2} cos⁡(\frac{3t}{2})+\frac{1}{2} sin⁡(\frac{3t}{2})\)

(d) \(\frac{1}{4} cos⁡(\frac{3t}{2})+\frac{1}{2} sin⁡(\frac{3t}{2})\)

I had been asked this question in exam.

My question is from General Properties of Inverse Laplace Transform in division Laplace Transform of Engineering Mathematics

Answer 1

Right answer is (d) \(\frac{1}{4} cos⁡(\frac{3t}{2})+\frac{1}{2} sin⁡(\frac{3t}{2})\)

Easy explanation: In the given question

=\(\frac{1}{4} L^{-1}\left (\frac{s+3}{s^2+\frac{9}{4}}\right )\)

=\(\frac{1}{4} \Big\{L^{-1}\left (\frac{s}{s^2+\frac{9}{4}}\right)+L^{-1}\left (\frac{3}{s^2+\frac{9}{4}}\right)\Big\}\)

=\(\frac{1}{4} \Big\{cos⁡(\frac{3t}{2})+2 sin⁡(\frac{3t}{2})\Big\}\)

=\(\frac{1}{4} cos⁡(\frac{3t}{2})+\frac{1}{2} sin⁡(\frac{3t}{2})\).