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بخش 3 این تمرینات رو برام حل کن

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

### Problem 1:
\[
\int_0^\infty \frac{e^{-ax} - e^{-bx}}{x} \, dx
\]

### 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:
\[
\mathcal{L}\{1\}(s) - \mathcal{L}\{1\}(s) = \int_0^\infty e^{-ax} \, dx - \int_0^\infty e^{-bx} \, dx
\]

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

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

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

### Problem 2:
\[
\int_0^\infty \frac{e^{-ax} \sin(bx)}{x} \, dx
\]

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

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

### Summary:
- For \(\int_0^\infty \frac{e^{-ax} - e^{-bx}}{x} \, dx\), the solution is \(\ln\left(\frac{b}{a}\right)\).
- For \(\int_0^\infty \frac{e^{-ax} \sin(bx)}{x} \, dx\), the solution is \(\arctan\left(\frac{b}{a}\right)\).

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.

Explore how to solve integrals involving exponential and sine functions using Laplace transforms. Detailed step-by-step solutions for each problem in section 3 exercises. Learn the applications of Laplace transforms in calculus and differential equations.

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