Logarithmic Integral Homework: Solving S [(e^x)(e^x)]/((2+(e^x))^2) dx

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Homework Help Overview

The discussion revolves around evaluating the integral of the function (e^(2x))/((2+(e^x))^2) with respect to x. The problem involves the use of substitution and integration techniques related to logarithmic functions.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to solve the integral by substituting u = 2 + (e^x) and expresses the integral in terms of u. They question how to proceed after substituting the expression for (e^x).
  • One participant suggests splitting the fraction to simplify the integration process, while another raises a concern about the integration of the resulting terms.
  • Further discussion includes clarifications on the integration of logarithmic functions and the importance of careful notation.

Discussion Status

The discussion is ongoing, with participants exploring different aspects of the integration process. Some guidance has been offered regarding the splitting of fractions and the integration of logarithmic terms, but there is no explicit consensus on the next steps.

Contextual Notes

Participants note the potential for confusion due to notation and the importance of careful attention to detail in mathematical expressions. There is an acknowledgment of the challenges posed by the integral's structure.

ppkjref
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Homework Statement


S = integral symbol
S (e^(2x))/((2+(e^x))^2) dx


Homework Equations


u = 2+(e^x)
(e^x) = u-2
du = (e^x) dx


The Attempt at a Solution


S [(e^x)(e^x)]/((2+(e^x))^2) dx
S [du(u-2)]/(u^2)
For the first (e^x) dx I substituted du. And since there was only (e^x) left, I substituted in u-2.
Now what do I do?
I know how to do du/(u^2). but what about the (u-2)?
 
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So you need to find \int \frac{u-2}{u^2} du . Just split the numerator like this: \frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}.
 
the a/c fraction is u/(u^2), equivalent to 1/u which when integrated is 0?
 
\int \frac{1}{u} du = \log_e u.

You should be familiar with that already.
 
ln(2+(e^x) + 2/(2+(e^x))
Just needed a break I guess. Wasn't thinking.
 
Last edited:
ppkjref said:
ln(2+(e^x)) + 2/(2+(e^x))
Just needed a break I guess. Wasn't thinking.

An unmatched left parentheses can haunt you all day.
 

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