How do I solve this integral using integration by parts?

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Homework Help Overview

The discussion revolves around solving the integral of the form (x^m)*(1-x)^k, where m is a nonnegative integer and k > -1. The original poster seeks assistance with integration by parts to evaluate this integral.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the nature of the integral, with one noting it relates to the incomplete beta function. There are questions about the expectations for the solution and the implications of the integral being definite versus indefinite. The original poster attempts to connect the integral to an induction proof.

Discussion Status

The conversation is ongoing, with participants exploring different interpretations of the integral and its properties. Some guidance has been offered regarding the use of integration by parts and the suggestion to evaluate specific cases of m to understand the pattern better.

Contextual Notes

There is mention of the integral being evaluated from 1 to 0, and the original poster indicates that the problem may involve induction, which adds complexity to the discussion. The constraints of the problem, such as the conditions on m and k, are acknowledged but not resolved.

peace-Econ
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Homework Statement



Solve the integral.

Homework Equations



Integral: (x^m)*(1-x)^k where m is a nonnegative integer and k > -1

The Attempt at a Solution



I've tried to take this integral by using integral by parts, but I couldn't take it. Can anyone tell me how to take this integral? I really appreciate that.
 
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If it's an indefinite integral then that's an incomplete beta function. You can do a lot of things with it, but you can't write a simple elementary form for it. What do they really want you to do?
 
Sorry. the integral is actually 1 to 0. This question is actually induction.

Integral(1-0): (x^m)*(1-x)^k=n!/(k+1)(k+2)...(K+m+1) where m is a nonnegative integer and k > -1

So, I thought that if I take integral from the right side, I can prove it. But it does not seem the case...
 
peace-Econ said:
Sorry. the integral is actually 1 to 0. This question is actually induction.

Integral(1-0): (x^m)*(1-x)^k=n!/(k+1)(k+2)...(K+m+1) where m is a nonnegative integer and k > -1

So, I thought that if I take integral from the right side, I can prove it. But it does not seem the case...

Then start working on the integration by parts idea. Call your integral I(m,k). Work out m=0. For practice try doing small values of m (m=1, m=2, m=3) until you see what's going on. Then try to express I(m+1,k) in terms of I(m,k) and apply induction.
 
Last edited:
actually, I think I made it. Thanks!
 

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