# Integration of an infinite product

1. Apr 5, 2013

### ecpietscheck

Hey guys, what sup
I need you all to help me in resolving the integral of an infinite product...
i was thinking of perhaps integrating by parts, but when yo do that the integration becomes brutally expansive...
any ideas?
thank you all very much

the variable which is aimed to be integrated is x btw...

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2. Apr 5, 2013

### pwsnafu

You wrote down a finite product. Do you in fact want $\int \prod_{m=1}^\infty (x^2 + m) \, dx$?

3. Apr 6, 2013

### kamke

for example

(x*x+1)*(x*x+2)*(x*x+3)*(x*x+4) = x^8 + a*x^6 + b*x^4 + c*x^2 + 1*2*3*4

1*2*3*4 = 4!

(x*x+1)*(x*x+2)*(x*x+3)*(x*x+4) = x^8 + 10x^6 + 35x^4 + 50x^2 + 24

Each member must be integrated.

∫ x^8 + 10x^6 + 35x^4 + 50x^2 + 24 dx =

= x^9/9 + 10x^7/7 + 7x^5 +50x^3/3 +24x + C

kamke

4. Apr 6, 2013

### Staff: Mentor

The problem is to get a general formula for those coefficients.

Basically $\displaystyle \sum_{i,j,...=1, i<j<...}^n i*j*k*...$ with 0 to n indices in the sum.
The second sum, for example, sums 1*2+1*3*...+1*n + 2*3+2*4+...+2*n+...

5. May 23, 2013

### ecpietscheck

MFB, that answer seems a bit unclear
perhaps any other ideas?

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