Prove Reduction Formula for Integral: x(1-x^3)^n dx

  • Thread starter Thread starter ani411
  • Start date Start date
  • Tags Tags
    Formula Reduction
Click For Summary

Homework Help Overview

The problem involves proving a reduction formula for the integral In = ∫ x(1-x^3)^n dx, with a specific relationship involving In and In-1. Participants are tasked with finding In in terms of n.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • One participant attempted integration by parts but encountered difficulties. Another participant questioned the correctness of the original statement regarding the integral. A third participant suggested a manipulation of the integral to express it in terms of I_n and I_(n-1).

Discussion Status

The discussion is ongoing, with various approaches being explored. Some participants are clarifying the problem statement while others are proposing different methods to tackle the integral. There is no explicit consensus on the correct approach yet.

Contextual Notes

Participants are navigating potential misunderstandings about the integral's formulation and the implications of the reduction formula. The original poster's attempt at a solution may not align with the expected proof, leading to further exploration of the problem.

ani411
Messages
2
Reaction score
0

Homework Statement



In= (the integral) x(1-x^3)^ndx

Prove that (3n +2)In = 3nIn-1 + x^2(1 - x)^n

Hence find In in terms of n

The Attempt at a Solution



I tried integration by parts (by letting u be (1-x^3)^n and got stuck after this:
In = 1/2x^2(1-x^3)^n + (the integral)(n(1-3x)^(n-1)(3x^2)(1/2x^2)dx)
= 1/2(x^2(1-x^3)^n +3n(the integral)(1-x^3)^(n-1)x^4dx

Would greatly appreciate any help and thanks in advance!
 
Physics news on Phys.org
I didn't find the result but I think you don't have the right thing to prove (the last x is an x^3).


(But I may be wrong).
 
Write I_n as integral of x*(1-x^3)^n=x*(1-x^3)*(1-x^3)^(n-1)=x(1-x^3)^(n-1)-x^4(1-x^3)^(n-1). You can express the pesky x^4(1-x^3)^(n-1) integral in terms of I_n and I_(n-1).
 
Ok, I get it now. Thanks(:
 

Similar threads

  • · Replies 10 ·
Replies
10
Views
2K
  • · Replies 1 ·
Replies
1
Views
2K
Replies
3
Views
2K
  • · Replies 105 ·
4
Replies
105
Views
10K
  • · Replies 5 ·
Replies
5
Views
3K
  • · Replies 14 ·
Replies
14
Views
2K
Replies
4
Views
3K
  • · Replies 16 ·
Replies
16
Views
3K
  • · Replies 7 ·
Replies
7
Views
3K
  • · Replies 5 ·
Replies
5
Views
2K