Math Problem Statement

بخش 3 این تمرینات رو برام حل کن

Solution

To solve the integrals in section 3 using Laplace transforms, we'll go through each problem step-by-step.

Problem 1:

0eaxebxxdx\int_0^\infty \frac{e^{-ax} - e^{-bx}}{x} \, dx

Solution:

We recognize that this integral can be evaluated using the properties of the Laplace transform.

The integral has the form of a Laplace transform difference: L{1}(s)L{1}(s)=0eaxdx0ebxdx\mathcal{L}\{1\}(s) - \mathcal{L}\{1\}(s) = \int_0^\infty e^{-ax} \, dx - \int_0^\infty e^{-bx} \, dx

Both integrals are improper and evaluated as: 0eaxdx=1a,0ebxdx=1b\int_0^\infty e^{-ax} \, dx = \frac{1}{a}, \quad \int_0^\infty e^{-bx} \, dx = \frac{1}{b}

However, given the form eaxebxx\frac{e^{-ax} - e^{-bx}}{x}, the integral can be transformed using: 0eaxebxxdx=ln(ba)fora,b>0\int_0^\infty \frac{e^{-ax} - e^{-bx}}{x} \, dx = \ln\left(\frac{b}{a}\right) \quad \text{for} \quad a, b > 0

Thus, the solution to the integral is: ln(ba)\ln\left(\frac{b}{a}\right)

Problem 2:

0eaxsin(bx)xdx\int_0^\infty \frac{e^{-ax} \sin(bx)}{x} \, dx

Solution:

This integral can be evaluated using the Laplace transform of the sine function combined with a known result: 0eaxsin(bx)xdx=arctan(ba)\int_0^\infty \frac{e^{-ax} \sin(bx)}{x} \, dx = \arctan\left(\frac{b}{a}\right)

Thus, the solution to the integral is: arctan(ba)\arctan\left(\frac{b}{a}\right)

Summary:

  • For 0eaxebxxdx\int_0^\infty \frac{e^{-ax} - e^{-bx}}{x} \, dx, the solution is ln(ba)\ln\left(\frac{b}{a}\right).
  • For 0eaxsin(bx)xdx\int_0^\infty \frac{e^{-ax} \sin(bx)}{x} \, dx, the solution is arctan(ba)\arctan\left(\frac{b}{a}\right).

Would you like detailed solutions for the other parts of the exercise or have any other questions? Here are some questions you might want to explore:

  1. How do you find the Laplace transform of a given function?
  2. What are the properties of the Laplace transform that simplify solving differential equations?
  3. Can you solve an inverse Laplace transform for a complex function?
  4. How are convolution integrals evaluated using the Laplace transform?
  5. What are some practical applications of the Laplace transform in engineering and physics?

Tip:

When solving integrals using Laplace transforms, always check if the integral fits a standard form or if it can be simplified using known properties of the Laplace transform.

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Math Problem Analysis

Mathematical Concepts

Integral Calculus
Laplace Transforms

Formulas

Laplace transform of exponential functions
Laplace transform of sine function

Theorems

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Suitable Grade Level

University Level