Integration of (e^x)dx/ (e^(2x) +5e^(x) + 6)

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Discussion Overview

The discussion revolves around the integration of the function (e^x)dx / (e^(2x) + 5e^(x) + 6). Participants explore different methods for solving the integral, including partial fraction decomposition and substitution techniques.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion regarding their partial fraction decomposition, stating they obtained A=3 and B=-2, but their derivative does not match the original function.
  • Another participant suggests using the substitution y=e^x as a potential approach to simplify the problem.
  • A later reply confirms the correctness of the partial fraction decomposition but emphasizes that the issue lies in the integration step.
  • Another participant advises to first substitute u=e^x, which leads to a simpler integration process before applying partial fractions.
  • Some participants argue that the order of substitution and partial fractions does not affect the final answer, while others contend that performing the substitution first is more straightforward.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to take for the integration. There are competing views on whether to substitute before or after applying partial fractions, and the discussion remains unresolved regarding the integration method.

Contextual Notes

Some participants note that the integration process may involve additional steps or decompositions that have not been fully detailed, leading to uncertainty in the solution.

k0hana27
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I was really confuse with this problem because after i simplify the denominator which is (e^x+2)(e^x+3) and use partial fraction i get A=3 and B=-2. Whenever I get the derivative of my answer it doens't match with the given (e^x)dx/ (e^(2x) +5e^(x) + 6). PLEASE help me :))
 
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Try using this way: Let y=ex :smile:
 
k0hana27 said:
I was really confuse with this problem because after i simplify the denominator which is (e^x+2)(e^x+3) and use partial fraction i get A=3 and B=-2. Whenever I get the derivative of my answer it doens't match with the given (e^x)dx/ (e^(2x) +5e^(x) + 6). PLEASE help me :))
Your partial fraction decomposition is certainly correct; so your problem lies in the following integration.

I suggest you use, from the outset, the substitution pointed out to you, and take the fractional decomposition afterwards.
 
You're rushing into partial fractions too quickly. First substitute u=e^x so that du=e^xdx. Partial fractions will then yield the correct integration.
 
It doesn't matter whether you substitute for e^x before or after the "partial fractions"- you would get the same answer. The real problem is that K0hana27 hasn't shown us how he integrated or what answer he got.
 
HallsofIvy said:
It doesn't matter whether you substitute for e^x before or after the "partial fractions"- you would get the same answer. The real problem is that K0hana27 hasn't shown us how he integrated or what answer he got.
Sure, it doesn't matter.

Except for the loathsome fact that if you afterwards performs the substitution, you'll most likely to do yet another partial fractions decomposition.
On both the two fractional terms previously gained.

It is easier to make the obvious substitution first.
 

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