Question

The Laplace transform of \( \frac{s}{s^2 + a^2} \) is:

• A. \( \cos(at) \)
• B. \( \sin(at) \)
• C. \( \sinh(at) \)
• D. \( \cosh(at) \)

Hint:

For Laplace transforms, remember that: \( \mathcal{L} \{ \cos(at) \} = \frac{s}{s^2 + a^2} \),
\( \mathcal{L} \{ \sin(at) \} = \frac{a}{s^2 + a^2} \).
These standard transforms help identify the inverse transforms more quickly.

Answer 1

The Correct Option is A

The Laplace transform of a function \( f(t) \) is defined as: \[ \mathcal{L} \{ f(t) \} = \int_0^\infty e^{-st} f(t) \, dt. \] We are asked to find the function whose Laplace transform is \( \frac{s}{s^2 + a^2} \). From standard Laplace transforms, we know that the Laplace transform of \( \cos(at) \) is: \[ \mathcal{L} \{ \cos(at) \} = \frac{s}{s^2 + a^2}. \] Thus, the given expression \( \frac{s}{s^2 + a^2} \) corresponds to the Laplace transform of \( \cos(at) \).