Integrating (1/(1+e^x)) dx: Challenges and Solutions

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SUMMARY

The integral of (1/(1+e^x)) dx can be effectively solved using substitution and partial fraction decomposition. The recommended substitution is u = 1 + e^x, leading to du = e^x dx, which simplifies the integral to ∫(du/((u-1)u)). An alternative method involves multiplying the integrand by e^-x, transforming the integral into ∫(e^-x/(e^-x + 1)) dx, and using the substitution u = e^-x + 1. Both methods ultimately require partial fraction decomposition for resolution.

PREREQUISITES
  • Understanding of integral calculus, specifically integration techniques.
  • Familiarity with substitution methods in integration.
  • Knowledge of partial fraction decomposition.
  • Basic proficiency with exponential functions and their properties.
NEXT STEPS
  • Study advanced integration techniques, focusing on substitution and partial fractions.
  • Practice solving integrals involving exponential functions, particularly ∫(1/(1+e^x)) dx.
  • Explore the application of logarithmic differentiation in integration problems.
  • Review calculus concepts related to limits and continuity to strengthen foundational knowledge.
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Students in calculus courses, mathematics educators, and anyone looking to deepen their understanding of integration techniques involving exponential functions.

  • #31
Are you differentiating \ln(u^2 -u) wrt "x" or to "u" ?
 
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  • #32
opps I meant to type d/du
 
  • #33
camilus said:
\int{1 \over u^2-u}du = ln(u^2-u) + C
Why would you think that?



p.s. have you looked at your private messages?
 
Last edited:
  • #35
Lol jesus how many mathematicians does it take to solve an exponential integral problem?

I'm not calling anyone stupid, but just found it funny ya'll are being cooperative to solve this
 
  • #36
Solve a man's integral problem, and he gets one answer. Teach a man how to solve integral problems, and he gets all of his homework done. :-p (and without cheating)
 
Last edited:
  • #37
Hurkyl said:
Solve a man's integral problem, and he gets one answer. Teach a man how to solve integral problems, and he gets all of his homework done. :-p (and without cheating)

(APPLAUSE) Well Said!
 

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