Evaluate each of the following definite integrals by using the Laplace transform table. $$\int_{0}^{\infty} t e^{-t} \sin 5 t d t$$

A definite integral is used to calculate the net area under a curve. It helps in finding the accumulation of quantities, such as areas, volumes, and other quantity integrations over intervals. In the problem, we compute the integral \( \int_{0}^{\infty} t e^{-t} \sin 5 t \ d t\ \). This confirms that we are calculating an accumulation from \( t = 0 \) to \( t = \infty \).\ Each term within the integral has a practical element:

• \( t \) could represent time mature.
• \( e^{-t} \) indicates an exponentially decaying factor.
• \( \sin 5t \) represents an oscillatory behavior.

Combining these in the definite integral seeks to understand how these elements interact over a continuous range.
Mathematically, it provides a finite value for such continuous processes. Definite integrals play a key role in fields like physics, economics, and engineering, enabling the solution of real-world problems. This accumulation is handled through Laplace transforms in this context.

The Laplace transform table is a powerful tool to convert complex time domain functions into simpler s-domain (complex frequency domain) functions. The table lists common functions and their corresponding Laplace transforms, significantly simplifying the solving process.
For example, in the exercise:

• The Laplace transform of \( e^{-t} \sin(5t) \) is \ \frac{5}{(s+1)^2 + 25} \ \ (from the table).

Using the table:

• We convert time domain functions like \( e^{-t} \sin(5t) \) into s-domain functions, easily manageable algebraically.
• We then utilize properties like differentiation and shifting to manipulate these transforms further.

When solving integrals, the Laplace transform table helps bypass complex direct integration, making it indispensable for engineers and scientists.

Differentiation is the process of finding the derivative of a function, showing the rate at which a function changes at any point. In the context of the Laplace transform: The exercise involves differentiating \( \aug^2 = (s+1)^2 + 25 \) with respect to \( s \).

Using the quotient rule for differentiation:

• We set \( u = 5 \) and \( v = (s+1)^2 + 25 \).
• \ \frac{d}{ds} \left( \frac{u}{v} \right) = \frac{v \ \frac{du}{ds} - u \ \frac{dv}{ds}}{v^2} \.

Applying this, we get:

• \frac{v \ \frac{du}{ds} - u \ \frac{dv}{ds}}{v^2} = \- \ \left( \frac{0 \ \times ( (s+1)^2 + 25) - 5 \ \times 2(s+1)}{( (s+1)^2 + 25)^2} \right)

Ultimately, this differentiation provides the simplified expression used to evaluate the integral. Differentiation is crucial in connections with modification, optimization, and evaluating functions' instantaneous rates of change, making it fundamental to calculus and applied mathematics.