Laplace transform if cos(at)u(t) is? a) s/a² s²? b) a/a² s² c) s²/a² s² d) a²/a² s²​

Break Down the Problem

1. We need to find the Laplace transform of the function $\cos(at)u(t)$ where $u(t)$ is the unit step function.
2. Identify the standard formula for the Laplace transform of the cosine function.

Relevant Concepts

• The Laplace transform of $f(t)$, denoted $L\{f(t)\}$, is given by: $L\{f(t)\} = \int_0^\infty e^{-st} f(t) \, dt$
• For the cosine function, the Laplace transform is: $L\{\cos(at)u(t)\} = \frac{s}{s^2 + a^2}$

Analysis and Detail

1. In our case, $f(t) = \cos(at)$ and it is multiplied by $u(t)$, which is 1 for $t \geq 0$.
2. Substituting into the Laplace transform formula, we calculate: $L\{\cos(at)u(t)\} = \int_0^\infty e^{-st} \cos(at) \, dt$ This integral evaluates to: $\frac{s}{s^2 + a^2}$

Verify and Summarize

• We have confirmed that the Laplace transform of $\cos(at)$ derived from the standard formula corresponds to what is computed.
• The resulting expression is consistent with our expectations and mathematical proofs.

Final Answer

The Laplace transform of $\cos(at)u(t)$ is: $\frac{s}{s^2 + a^2}$ Hence, none of the provided answer choices (a, b, c, d) are correct.