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

[ani411]

Homework Statement

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

Prove that (3n +2)I_n = 3nI_n-1 + x^2(1 - x)^n

Hence find I_n 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:
I_n = 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!

 

[penguin007]

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).

 

[Dick]

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).

 

[ani411]

Ok, I get it now. Thanks(: