Integral using Euler's formula.

In summary, the conversation discusses the use of Euler's formula to simplify the integration of e^{2x}sin(x)sin(2x). However, the use of Euler's formula did not give the correct answer. The conversation then explores different approaches to solving the problem, including manipulating the expression into a form with only exponentials on the left side and using trigonometric identities to simplify the integrals. Finally, the conversation concludes with a suggestion to use the identity sina*sinb = -0.5[cos(a+b)+cos(a-b)] to solve the integrals.
  • #1
cragar
2,552
3
If I have [itex] \int e^{2x}sin(x)sin(2x) [/itex]
And then I use Eulers formula to substitute in for the sine terms.
So I have [itex] \int e^{2x}e^{ix}e^{2ix} [/itex]
then I combine everything so i get
[itex] e^{(2+3i)x} [/itex]
so then we integrate the exponential, so we divide by 2+3i
and then i multiply by the complex conjugate. now since sine is the imaginary part of his
formula I took the imaginary part when I back substituted for e^(3i)
but I didn't get the correct answer doing this, so am i not using Eulers formula correctly?
 
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  • #2
e^i3x = sin3x+isin2x , so the imaginary part is different from i(sinxsin2x)
 
  • #3
why does e^i3x = sin3x+isin2x , i guess I am not seeing it off hand I probably should look at it more and try to manipulate it more.
 
  • #4
sry typos , its sin3x
 
  • #5
how come one part is not cos(3x)
 
  • #6
another typos , sry =='
 
  • #7
But we could get it in the form of
[itex] sin(x)e^{2ix}=isin(2x)sin(x)+cos(2x)sin(x) [/itex]
Do we need to get an expression where we have just exponentials on the left hand side
and then isin(x)sin(2x)+cos(2x)cos(x)
 
  • #8
but then ur integral can't become e^i3x now , can it
 
  • #9
ok, I am not sure exactly what you mean, How do you recommend I approach the problem.
 
  • #10
sina*sinb = -0.5[cos(a+b)+cos(a-b)] , then u have 2 solvable integrals
 
  • #11
oh i see thanks for your answer.
 

1. What is Euler's formula and how is it used in integrals?

Euler's formula is a mathematical equation that relates the complex exponential function to the trigonometric functions. In integrals, it is used to convert trigonometric functions into exponential functions, making them easier to solve.

2. What is the general form of an integral using Euler's formula?

The general form of an integral using Euler's formula is ∫a to b f(x) cos(nx) dx = (1/2π) ∫a to b f(x) e^(inx) + f(x) e^(-inx) dx, where n is a positive integer.

3. How does Euler's formula simplify the process of solving integrals?

Euler's formula simplifies the process of solving integrals by converting trigonometric functions into exponential functions. This makes the integrals easier to evaluate using basic integration techniques, such as substitution and integration by parts.

4. Are there any limitations to using Euler's formula in integrals?

Yes, there are limitations to using Euler's formula in integrals. It can only be used for integrals involving trigonometric functions, and it may not work for all trigonometric functions. Additionally, it may not be the most efficient method for evaluating certain integrals.

5. Can Euler's formula be used in higher dimensions for multiple integrals?

Yes, Euler's formula can be extended to higher dimensions for multiple integrals. In this case, the formula becomes ∫∫...∫ f(x1, x2, ..., xn) cos(n1x1 + n2x2 + ... + nnxn) dx1dx2...dxn = (1/2π)^n ∫∫...∫ f(x1, x2, ..., xn) e^(i(n1x1 + n2x2 + ... + nnxn)) dx1dx2...dxn, where n1, n2, ..., nn are positive integers.

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