Integration by Parts with Complex Exponentials

In summary, Integration by parts with complex exponentials is a method used to integrate functions involving complex exponentials. It is particularly useful for integrals that involve products of exponential and trigonometric functions and cannot be solved using other methods. The steps involved include identifying the parts of the integral, using the product rule of differentiation, and checking for accuracy. It is important to choose the right functions to differentiate and integrate, and to simplify the integral before applying this method. However, it may not be applicable for all types of integrals.
  • #1
abney317
1
0
I'm looking at this problem here. (Exam practice, move to homework if you want...)
nZ4jRd1.png


First part is easy, but it's the second part that I can't quite figure out.


I'm trying to get from ∫xe2x cos(2x)dx to this answer:
¼e2x(x cos(2x) + x sin(2x) - ½sin(2x))+C
 
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  • #2
Do you know Euler's formula?
 
  • #3
[tex]cos(2x)= \frac{e^{2ix}+ e^{-2ix}}{2}[/tex]
so
[tex]\int xe^{2x} cos(2x)dx= \frac{1}{2}\int x(e^{2(1+ i)x}+ e^{2(1- i)x})dx[/tex]
 

1. What is integration by parts with complex exponentials?

Integration by parts with complex exponentials is a method used to integrate functions involving complex exponentials. It involves breaking down the original integral into two parts and using the product rule of differentiation to simplify the integral. This method is particularly useful when dealing with integrals that involve products of exponential and trigonometric functions.

2. How is integration by parts with complex exponentials used?

Integration by parts with complex exponentials is used to simplify integrals involving complex exponentials. It is particularly useful in solving integrals that cannot be solved using other methods, such as substitution or partial fractions. It can also be used to solve certain differential equations.

3. What are the steps involved in integration by parts with complex exponentials?

The first step in integration by parts with complex exponentials is to identify the parts of the integral that can be broken down into two factors. Then, using the product rule of differentiation, one of the factors is differentiated while the other is integrated. This process is repeated until the integral can be solved. Finally, the solution is checked for accuracy.

4. What are some tips for solving integrals using integration by parts with complex exponentials?

When solving integrals using integration by parts with complex exponentials, it is important to choose the right functions to differentiate and integrate. Generally, choosing the function with the more complicated derivative and the simpler integral will make the integration process easier. It is also helpful to simplify the integral as much as possible before applying the integration by parts method.

5. Can integration by parts with complex exponentials be used for all integrals?

No, integration by parts with complex exponentials may not be applicable for all integrals. It is best suited for integrals involving products of exponential and trigonometric functions. For other types of integrals, other methods such as substitution or partial fractions may be more appropriate.

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