Integration of the binomial theorem

[hmmmmm]

is it possible to integrate the binomial theorem??

 

[tiny-tim]

Welcome to PF!

Hiho hmmmmm! Welcome to PF!

is it possible to integrate the binomial theorem??

Yes, and that should give you an equation relating ⁿC_k and ^n-1C_k-1 …

what is it?

 

[HallsofIvy]

What do you mean by "integrating a theorem", integrating both sides of the equation involved? If so, yes, tiny-tim is correct.

 

[hmmmmm]

ok so what would happen if you integrated both sides of it then.

 

[tiny-tim]

ok so what would happen if you integrated both sides of it then.

uh-uh … you tell us!

 

[HallsofIvy]

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?

 

[hmmmmm]

yes this is the part i don't understand how you would integrate (a^x+B)^y once expanded not in the braket.

 

[HallsofIvy]

"(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.

 

[hmmmmm]

i mean the integral of (a^x+b)^y da where x b and y are constants sorry for my sloppy notation

 

[HallsofIvy]

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.