How to Solve for the Integral of x^(n-1) from 0 to 1?

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Homework Help Overview

The discussion revolves around finding the integral of the function x^(n-1) from 0 to 1, which falls under the subject area of calculus, specifically integral calculus.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants explore the use of induction to establish a recurrence relation for the integral. There are attempts to manipulate the integral using algebraic techniques, and questions arise regarding the transition to a specific form involving 1/n.

Discussion Status

The discussion is ongoing, with participants providing insights into potential methods for solving the integral. Some guidance has been offered regarding the use of induction and algebraic manipulation, but there is no explicit consensus on the final approach or solution.

Contextual Notes

Participants are navigating through the complexities of the integral and its properties, with some expressing uncertainty about specific algebraic steps and the implications of the recurrence relation.

gomes.
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Having a bit of trouble, any help will be appreciated.
Thanks.
 
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gomes. said:
Having a bit of trouble, any help will be appreciated.
Thanks.

Looks like you'll probably have to use induction on n.

1. Show that the recurrence relation is true for n = 1. (base step)
2. Assume that, for some n in the natural numbers, your recurrence relation holds. Try to show that n + 1 holds as well. (inductive step).

It's probably going to involve a bit of messy algebra.
 
I_{n+1} = \int_0^1 \frac{x^n}{2-x} dx = \int_0^1 \frac{x^{n-1}(2 - (2-x))}{2-x} dx = 2\int_0^1 \frac{x^{n-1}}{2-x} dx - \int_0^1 x^{n-1} dx

= ...
 
Deadstar said:
\int_0^1 x^{n-1} dx

= ...

Thanks, I've got it now (the 2I(n) ) bit, but how do i get the above part to 1/n?
 

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