Final answer:
To verify the Laplace transform L[sin(bt)] = b/s²+b², we apply the definition of the Laplace transform and use a trigonometric identity to rewrite the function. Then, we integrate the result using the definition of the Laplace transform.
Step-by-step explanation:
To verify the Laplace transform L[sin(bt)] = b/s²+b², we need to use the definition of the Laplace transform. The definition of the Laplace transform states that L[f(t)] = ∫[0, ∞] f(t)e^(-st) dt, where L[f(t)] is the Laplace transform of f(t), s is the Laplace variable, and t is the time variable.
Applying the definition of the Laplace transform to L[sin(bt)], we have:
L[sin(bt)] = ∫[0, ∞] sin(bt)e^(-st) dt
Next, we can use the trigonometric identity sin(A + B) = sin(A)cos(B) + cos(A)sin(B) to rewrite sin(bt) as sin(0 + bt). Applying this identity, we get:
L[sin(bt)] = ∫[0, ∞] sin(0 + bt)e^(-st) dt