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

In summary, the conversation is about proving the equation (3n + 2)I_n = 3nI_(n-1) + x^2(1-x)^n, and finding I_n in terms of n. The attempt at a solution involved using integration by parts, but the correct result was not obtained. The expert suggests writing I_n in terms of x*(1-x^3)^n and using the equation to simplify the integral.
  • #1
ani411
2
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!
 
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  • #2
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).
 
  • #3
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).
 
  • #4
Ok, I get it now. Thanks(:
 

1. What is the reduction formula for the given integral?

The reduction formula for the given integral is: In = (1/(4n+1)) * [(1-x3)n+1 - (1-x3)n] + C, where C is the constant of integration.

2. How is the reduction formula derived?

The reduction formula is derived using the technique of integration by parts. By choosing u = x and dv = (1-x3)n dx, we get du = dx and v = (1/(4n+1)) * (1-x3)n+1. Substituting these values in the integration by parts formula, we get the reduction formula.

3. What is the purpose of using the reduction formula?

The reduction formula is used to simplify the integration process of a given integral. By using the formula, we can reduce the power of the polynomial in the integrand, making it easier to integrate.

4. Can the reduction formula be used for any value of n?

Yes, the reduction formula can be used for any value of n, as long as it is a positive integer. It is a general formula that is applicable to all values of n.

5. How can the reduction formula be utilized in solving definite integrals?

The reduction formula can be used to solve definite integrals by repeatedly applying the formula and substituting the limits of integration in each step. This will eventually lead to the solution of the definite integral in terms of n and the constant of integration C.

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