Laplace transform of 𝐿𝑠𝑖𝑛𝑎𝑡Lsinat =a)𝑎2𝑠2−𝑎2s 2 −a 2 a 2 ​ b)𝑎𝑠2+𝑎2s 2 +a 2 a​ c)𝑠𝑠2−𝑎2s 2 −a 2 s​ d)𝑎𝑠2−𝑎2s 2 −a 2 a​

To find the Laplace transform of $L\sin(at)$, we can follow these steps:

1. Break Down the Problem

We need to compute the Laplace transform of the function $\sin(at)$. The Laplace transform is defined as:

$L\{f(t)\} = \int_0^{\infty} e^{-st} f(t) \, dt$

In our case, $f(t) = \sin(at)$.

2. Relevant Concepts

The Laplace transform of $\sin(at)$ is given by the formula:

$L\{\sin(at)\} = \frac{a}{s^2 + a^2}$

This formula will help us arrive at the correct answer.

3. Analysis and Detail

Now we apply the Laplace transform formula directly for $\sin(at)$:

• Substitute $a$ for the frequency of the sine function in the formula.
• We obtain:

$L\{\sin(at)\} = \frac{a}{s^2 + a^2}$

4. Verify and Summarize

We can check the standard result of the Laplace transform for correctness. Recalling the derivation confirms that our calculation aligns with established knowledge in Laplace transforms.

Final Answer

The correct answer for the Laplace transform $L\sin(at)$ is:

$\frac{a}{s^2 + a^2}$

Based on the options given, none seem to represent the answer directly. However, the Laplace transform $L\{\sin(at)\} = \frac{a}{s^2 + a^2}$ is correct.