https://file.aiquickdraw.com/mathbot/file/1728776564200idesjkm0.jpg

The expression in the image appears to involve the Laplace transform of a function involving \(\sin(ax)\). The notation suggests:

\[
\mathcal{L}\{ \sin(ax) \} = \int_0^{\infty} e^{-sx} \sin(ax) \, dx
\]

This is the definition of the Laplace transform of the sine function, where \(\mathcal{L}\) represents the Laplace transform, and \(s\) is the complex frequency parameter.

### Solving this Laplace transform:

\[
\mathcal{L}\{\sin(ax)\} = \int_0^{\infty} e^{-sx} \sin(ax) \, dx
\]

We can solve this integral using integration by parts or by consulting standard Laplace transform tables.

The standard result for this transform is:

\[
\mathcal{L}\{\sin(ax)\} = \frac{a}{s^2 + a^2}
\]

Thus, the Laplace transform of \(\sin(ax)\) is:

\[
\int_0^{\infty} e^{-sx} \sin(ax) \, dx = \frac{a}{s^2 + a^2}
\]

Would you like further details on the steps of solving this integral?

### Related Questions:
1. How does the Laplace transform of \(\cos(ax)\) differ from \(\sin(ax)\)?
2. What is the inverse Laplace transform of \(\frac{a}{s^2 + a^2}\)?
3. How can we apply the Laplace transform to solve differential equations?
4. What is the physical interpretation of the Laplace transform in control theory?
5. How would the Laplace transform change if the sine function had a different phase shift?

### Tip:
The Laplace transform is a powerful tool to convert differential equations into algebraic equations, making them easier to solve.

Understand how to find the Laplace transform of sin(ax). The solution involves key concepts in integral calculus and the properties of Laplace transforms. The result is L{sin(ax)} = a / (s^2 + a^2). Suitable for undergraduate students studying math or engineering.

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