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integrate x^3/(exp(x)-1) between 0 and infinity

If I type

integrate(x^3/(exp(x)-1),x,0,infinity)

I get

-1/15pi^4 + limit(-1/4x^4 + x^3log(-e^x + 1) + 3x^2dilog(e^x) - 6xpolylog(3, e^x) + 6polylog(4, e^x), x, +Infinity, minus)

I have two questions :

  1. How do I evaluate the limit ?
  2. The correct answer is pi^4/15 : why SageMath does not give it ?

Thanks

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updated 2017-06-08 13:01:49 -0500

kcrisman gravatar image

integrate x^3/(exp(x)-1) between 0 and infinity

If I type

integrate(x^3/(exp(x)-1),x,0,infinity)

> integrate(x^3/(exp(x)-1),x,0,infinity)

I get

-1/15pi^4

-1/15*pi^4 + limit(-1/4x^4 limit(-1/4*x^4 + x^3log(-e^x x^3*log(-e^x + 1) + 3x^2dilog(e^x) 3*x^2*dilog(e^x) - 6xpolylog(3, 6*x*polylog(3, e^x) + 6polylog(4, 6*polylog(4, e^x), x, +Infinity, minus)

minus)

I have two questions :

  1. How do I evaluate the limit ?
  2. The correct answer is pi^4/15 : why SageMath does not give it ?

Thanks

integrate x^3/(exp(x)-1) between 0 and infinity

If I type

> 

integrate(x^3/(exp(x)-1),x,0,infinity)

I get -1/15pi^4 + limit(-1/4x^4 + x^3log(-e^x + 1) + 3x^2dilog(e^x) - 6xpolylog(3, e^x) + 6polylog(4, e^x), x, +Infinity, minus)

I getnumerical_integral(x^3/(exp(x)-1),0,infinity) gives 6.4939394075

-1/15*pi^4 + limit(-1/4*x^4 + x^3*log(-e^x + 1) + 3*x^2*dilog(e^x) - 6*x*polylog(3, e^x) + 6*polylog(4, e^x), x, +Infinity, minus)

I have two questions :

  1. How do I evaluate the limit ?
  2. The correct answer is pi^4/15 pi^4/15(=6.49393940226683) : why SageMath does not give it ?

Thanks

integrate x^3/(exp(x)-1) between 0 and infinity

If I type

integrate(x^3/(exp(x)-1),x,0,infinity) I get -1/15pi^4 + limit(-1/4x^4 + x^3log(-e^x + 1) + 3x^2dilog(e^x) - 6xpolylog(3, e^x) + 6polylog(4, e^x), x, +Infinity, minus)

numerical_integral(x^3/(exp(x)-1),0,infinity) gives 6.4939394075

I have two questions :

  1. How do I evaluate the limit ?
  2. The correct answer is pi^4/15(=6.49393940226683) : why SageMath does not give it ?

Thanks

integrate x^3/(exp(x)-1) between 0 and infinity

If I type

integrate(x^3/(exp(x)-1),x,0,infinity) I get -1/15pi^4 + limit(-1/4x^4 + x^3log(-e^x + 1) + 3x^2dilog(e^x) - 6xpolylog(3, e^x) + 6polylog(4, e^x), x, +Infinity, minus)

numerical_integral(x^3/(exp(x)-1),0,infinity) gives 6.4939394075

I have two questions :

  1. How do I evaluate the limit ?
  2. The correct answer is pi^4/15(=6.49393940226683) : why SageMath does not give it ?

Thanks

integrate x^3/(exp(x)-1) between 0 and infinity

If I type

integrate(x^3/(exp(x)-1),x,0,infinity) I get -1/15pi^4 + limit(-1/4x^4 + x^3log(-e^x + 1) + 3x^2dilog(e^x) - 6xpolylog(3, e^x) + 6polylog(4, e^x), x, +Infinity, minus)

The command numerical_integral(x^3/(exp(x)-1),0,infinity) gives 6.4939394075

I have two questions :

  1. How do I evaluate the limit ?
  2. The correct answer is pi^4/15(=6.49393940226683) : why SageMath does not give it with symbolic integration ?

Thanks