How to integrate e^x cos(x) using parts?

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SUMMARY

The integration of the function e^x cos(x) can be achieved using integration by parts, specifically by applying the technique twice. This results in a circular equation where the integral I can be expressed as I = something - I, leading to the conclusion I = something/2. Additionally, the problem can be approached using de Moivre's theorem, which states that cos(x) + j*sin(x) = e^(j*x).

PREREQUISITES
  • Understanding of integration techniques, specifically integration by parts.
  • Familiarity with complex numbers and de Moivre's theorem.
  • Knowledge of exponential functions and their properties.
  • Basic calculus concepts, including definite and indefinite integrals.
NEXT STEPS
  • Practice integration by parts with various functions to solidify understanding.
  • Study de Moivre's theorem and its applications in complex analysis.
  • Explore the properties of exponential functions in calculus.
  • Learn about the relationship between trigonometric functions and complex exponentials.
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Students and educators in calculus, mathematicians exploring integration techniques, and anyone interested in the application of complex numbers in integration.

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Is it possible to intergrate

e^x (cosx)

i wondered because i tried to intergrate it by parts, but ended up going round in circles.

I wondered because i had this question and I am stuck on how to do it :)

http://img505.imageshack.us/img505/320/frfbc8.png
 
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Sure it's possible. Just integrate by parts twice. You will get back to where you started - but if you call the integral I, then you will get I=somthing-I. So I=something/2. Not circular. On the other hand the question you posted is done more directly with deMoivre. cos(x)+j*sin(x)=e^(j*x).
 

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