The Laplace Transform is a mathematical tool used to solve differential equations. It transforms a function of time into a function of complex frequency, making it easier to solve equations involving derivatives.
To solve a Laplace Transform, you must first take the function of time and multiply it by the exponential function e^-st. Then, integrate the resulting function from 0 to infinity and plug in the value of s. This will give you the transformed function in terms of complex frequency.
The integral in the given equation is used to solve for the Laplace Transform of a function. It is a key step in the process, as it transforms the function of time into a function of complex frequency.
The sine and cosine functions represent the frequency of the original function. In this case, the frequency is 9.
The Laplace Transform has many applications in engineering, physics, and other fields. It can be used to solve differential equations that model real-world systems, such as electrical circuits, mechanical systems, and chemical reactions. It is also used in signal processing and control systems.