In exercises the laplace transform of a function f is the

Question

Exercise -

In Exercises, the Laplace transform of a function f is the function L f(s) of the variable s defined by the improper integral (if it converges):

Lf(s) = ₀∫^∞f(x)e^-sx dx

Laplace transforms are widely used in physics and engineering.

1. Show that if f (x) = C, where C is a constant, then Lf(s) = C/s for s > 0.

2. Show that if f (x) = sin αx, then Lf(s) = α/(s²+α²).

3. Compute Lf(s), where f(x) = e^αx and s > α.

4. Compute Lf(s), where f(x) = cos αx and s > 0.