Chapter 15: Problem 38

Evaluate each of the following definite integrals by using the Laplace transform table. $$ \int_{0}^{\pi} e^{-t}(1-\cos 2 t) d t $$

Short Answer

Expert verified

Laplace integrals involve seeing boundaries with transforms and evaluating initial expressions.

Step by step solution

Identify the Laplace Transform

Recognize the integral's form and look for the Laplace transforms of the components in the integral. Here we need the transforms for $e^{-t}$ and $e^{-t}\text{cos}(2t)$.

Find the Laplace Transform of Individual Functions

Consult a Laplace transform table. We find that:1. $ \text{Laplace}(e^{-t}) = \frac{1}{s+1} $2. $ \text{Laplace}(e^{-t}\text{cos}(2t)) = \frac{s+1}{(s+1)^2 + 4} $

Linearity Property of the Laplace Transform

Use the linearity property of the Laplace transform, which states that:$ \text{Laplace}(af(t) + bg(t)) = a \text{Laplace}(f(t)) + b \text{Laplace}(g(t)) $Here, the integral becomes a combination of the transforms found.

Apply the Bounds to Find the Integral

Evaluate the transform from 0 to $ \frac{\pi}{s+1} \ $:$\int_0^{\pi} e^{-t}(1-\text{cos}(2t)) dt = \text{Laplace}^{-1}\bigg(\frac{1}{s+1}\bigg|_0^{\pi} - \frac{s+1}{(s+1)^2 + 4}\bigg|_0^{\pi}\bigg) $

Evaluate Inverse Transforms

Use the inverse transform at the bounds and subtract the lower limit from the upper limit values. Simplify results as much as needed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

definite integrals

Definite integrals are a fundamental concept in calculus. They represent the area under the curve of a function between two specified bounds, say from $ a $ to $ b $. In our context, we are evaluating the definite integral \[ \int_{0}^{\pi} e^{-t}(1 - \cos(2t)) dt \]. This specific type of integral can often be tackled using advanced techniques like Laplace transforms, especially when dealing with complex exponentials and trigonometric functions. When handling definite integrals, remember:

The integral calculates the net area between the curve and the x-axis over the interval.
Integrals can represent a wide variety of physical quantities such as distance, area, volume, and more.

Understanding these basics can make it easier to apply tools like the Laplace transform.

Laplace transform table

The Laplace transform table is an invaluable tool for quickly identifying the Laplace transforms of common functions. To solve the integral, we used the transform table to find the transforms of $ e^{-t} $ and $ e^{-t} \cos(2t) $. The pertinent transforms are:

$ \text{Laplace}(e^{-t}) = \frac{1}{s+1} $
$ \text{Laplace}(e^{-t} \cos(2t)) = \frac{s+1}{(s+1)^2 + 4} $

Using these transformations allows us to move our problem from the time domain into the s-domain (or Laplace domain). This makes it easier to manipulate and eventually solve the integral. The table saves time and avoids the need for lengthy calculations by hand, acting as a reference for quick lookup.

linearity property of Laplace transforms

The linearity property of Laplace transforms is a powerful attribute. It states that the Laplace transform of a linear combination of functions is the same combination of their individual transforms. Mathematically:

$ \text{Laplace}(af(t) + bg(t)) = a \text{Laplace}(f(t)) + b \text{Laplace}(g(t)) $

This property simplifies the process of finding the Laplace transform of more complex functions by breaking them down into simpler components. In our integral, it helped us manage the combination of $ e^{-t} $ and $ \cos(2t) $ effectively. Understanding and applying the linearity property allows us to decompose problems and handle each part independently before recomposing the solution.

inverse Laplace transforms

To return to the time domain after using Laplace transforms, we use the concept of the inverse Laplace transform. This process transforms the function back from the s-domain to the time domain. In our example, after transforming each component, we apply the bounds and then take the inverse Laplace transform to solve the original definite integral. The steps are:

Apply the transforms and combine based on the integral's formula.
Evaluate within the given bounds from $ t = 0 $ to $ t = \pi $.
Perform the inverse Laplace transform to interpret the solution in the original domain.

This methodical approach ensures that the complex integration becomes more manageable and is executed accurately.