Integration of the binomial theorem

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Discussion Overview

The discussion revolves around the integration of the binomial theorem, exploring whether it is feasible to integrate expressions derived from the theorem and the implications of such integration. Participants engage with both theoretical and practical aspects of integration related to binomial expressions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants question the meaning of "integrating a theorem" and whether it involves integrating both sides of an equation.
  • One participant suggests that integrating both sides of the binomial theorem could yield a relationship between binomial coefficients.
  • Another participant proposes writing down the binomial theorem and integrating each side separately using the power rule.
  • Concerns are raised about integrating expressions like (a^x + B)^y, with a suggestion to use substitution for integration.
  • A participant clarifies their intent to integrate (a^x + b)^y with respect to 'a', acknowledging previous notation confusion.
  • One participant encourages the use of the binomial expansion to integrate term by term.

Areas of Agreement / Disagreement

Participants express varying interpretations of how to approach the integration of the binomial theorem, with no consensus on a single method or outcome. Multiple viewpoints on the integration process remain present.

Contextual Notes

Some participants highlight the need for clarity in notation and assumptions regarding constants in the expressions being integrated. There is also an emphasis on the importance of not expanding certain forms before integration.

Who May Find This Useful

This discussion may be useful for individuals interested in mathematical integration techniques, particularly in relation to the binomial theorem and its applications.

hmmmmm
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is it possible to integrate the binomial theorem??
 
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Welcome to PF!

Hiho hmmmmm! Welcome to PF! :smile:
hmmmmm said:
is it possible to integrate the binomial theorem??

Yes, and that should give you an equation relating nCk and n-1Ck-1 …

what is it? :smile:
 
What do you mean by "integrating a theorem", integrating both sides of the equation involved? If so, yes, tiny-tim is correct.
 
ok so what would happen if you integrated both sides of it then.
 
hmmmmm said:
ok so what would happen if you integrated both sides of it then.

uh-uh … you tell us! :smile:
 
1) Write down the formula for "the binomial theorem."

2) Integrate each side separately. That shouldn't be difficult, they both involve just the "power rule", that the integral of xn is (1/n+1)xn+1+ a constant.

What do you get?
 
yes this is the part i don't understand how you would integrate (a^x+B)^y once expanded not in the braket.
 
"(a^x+ B)^y is NOT a binomial form. Do you mean (Ax+ B)^y where A, B, and y are constants? If so, do not expand it. Make the substitution u= Ax+ B so that du= Adx and dx= (1/A)du.
 
i mean the integral of (a^x+b)^y da where x b and y are constants sorry for my sloppy notation
 
  • #10
Good! So do what we have been encouraging you do to all along! Write out (a^x+ b)^y in terms of the binomial expansion and integrate, term by term.
 

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