Cyclic Functions and Integration by Parts: Where Did I Go Wrong?

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Discussion Overview

The discussion revolves around the challenges of integrating cyclic functions, specifically the integral of (sinx)e^(inx), using integration by parts. Participants explore various methods and approaches to tackle these integrals, including the order of integration and the implications of repeated integration by parts.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion over obtaining a trivial result (C=C) when applying integration by parts to cyclic functions like (sinx)e^(inx), questioning whether they made an error or if alternative methods exist.
  • Another participant requests clarification and suggests that integrals typically resolve nicely when integration by parts is applied twice, hinting at the importance of collecting like terms.
  • A different participant advises stopping the use of integration by parts after obtaining the integral of the original function, suggesting to merge it with the original function and divide by a new constant to avoid trivial results.
  • One participant shares their experience of changing the order of integration, noting that it led to a successful outcome, indicating that the order may significantly affect the result.
  • Another participant elaborates on the process of integration by parts, explaining that reversing the roles of u and dv in a second application will simply cancel out the previous steps, leading to no new information.
  • A later reply acknowledges a mistake in understanding, indicating that the participant realized they had made the same error pointed out by another contributor.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to integrating cyclic functions using integration by parts. Multiple competing views and methods are presented, with some participants suggesting different strategies and others highlighting potential pitfalls.

Contextual Notes

Some limitations in the discussion include the dependence on the order of integration and the assumptions made about the functions involved. The discussion does not resolve the mathematical steps or the implications of the different approaches presented.

pivoxa15
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Why is it that when I do integration by parts on cyclic functions such as (sinx)e^(inx), I get a trivial answer like C=C, C is a constant

Have I done something wrong or are there other methods of doing those integrals?
 
Last edited:
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I don't understand your question, could you give a better example of what you mean? Integrals like this usually work out nicely by using parts twice and then collecting like terms of the original integral on one side of the equation (I'm not sure how much sense that made if any).
 
As soon as you get the integral of your original function, stop using parts. Merge it with the original function on the left and divide by the new constant. Using parts again just proves that it is equal (c=c)
 
I did the integral originally posted with a different order, exponetial on the left and it worked out nicely. The first time I did it, I had sin on the left and it turned into the trivial C=C (following all the usual rules). Looks like the order of integration matters.
 
If, integrating something like [itex]\int f(x)g(x)dx[/itex], you do an integration by parts, letting u= f(x), dv= g(x)dx, you will, of course, get another integral [itex]\int vdu[/itex]. If you do another integration by parts, this time reversing the roles (letting u= the old v and dv= the old u dx) you will just reverse what you did in the first integration and, yes, everything will cancel.
 
HallsofIvy said:
If, integrating something like [itex]\int f(x)g(x)dx[/itex], you do an integration by parts, letting u= f(x), dv= g(x)dx, you will, of course, get another integral [itex]\int vdu[/itex]. If you do another integration by parts, this time reversing the roles (letting u= the old v and dv= the old u dx) you will just reverse what you did in the first integration and, yes, everything will cancel.

I see what you are getting at here, I realized that I made the exact mistake you pointed out here.
 

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