- #1
hmmmmm
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is it possible to integrate the binomial theorem??
Integration of the binomial theorem
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In summary, the conversation discusses the possibility of integrating the binomial theorem, with a focus on (a^x+ b)^y as a specific example. It is suggested to write out the formula and integrate each side separately using the power rule. The conversation also includes clarification on the notation and suggests making a substitution to simplify the integration process.
- #2
tiny-tim
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Welcome to PF!
Hiho hmmmmm! Welcome to PF!
Yes, and that should give you an equation relating nCk and n-1Ck-1 …
what is it?
Hiho hmmmmm! Welcome to PF!
hmmmmm said:
Yes, and that should give you an equation relating nCk and n-1Ck-1 …
what is it?
- #3
HallsofIvy
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What do you mean by "integrating a theorem", integrating both sides of the equation involved? If so, yes, tiny-tim is correct.
- #4
hmmmmm
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ok so what would happen if you integrated both sides of it then.
- #5
tiny-tim
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hmmmmm said:
uh-uh … you tell us!
- #6
HallsofIvy
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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?
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?
- #7
hmmmmm
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yes this is the part i don't understand how you would integrate (a^x+B)^y once expanded not in the braket.
- #8
HallsofIvy
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"(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.
- #9
hmmmmm
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i mean the integral of (a^x+b)^y da where x b and y are constants sorry for my sloppy notation
- #10
HallsofIvy
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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.
FAQ: Integration of the binomial theorem
What is the binomial theorem and why is it important in integration?
The binomial theorem is a mathematical formula that allows us to expand expressions with binomial coefficients, typically used in algebra. It is important in integration because it helps us to simplify complex expressions and solve integration problems efficiently.
What are the steps involved in integrating using the binomial theorem?
The steps involved in integrating using the binomial theorem are: 1) Expanding the expression using the binomial theorem, 2) Simplifying the resulting expression, 3) Integrating each term separately, 4) Combining the integrals to get the final answer.
Can the binomial theorem be used for all types of integration problems?
No, the binomial theorem is mainly used for integrating polynomials and power functions. It may not be applicable for all types of integration problems, such as trigonometric, exponential, or logarithmic functions.
How is the binomial theorem related to the power rule in integration?
The binomial theorem can be seen as an extension of the power rule in integration. It allows us to integrate expressions with a variable raised to a power, but also includes additional terms with binomial coefficients.
Are there any limitations or restrictions when using the binomial theorem in integration?
One limitation of the binomial theorem is that it can only be used for functions with a finite number of terms. It may also not be applicable for functions with complicated or irrational exponents.