Black Body Radiation (Awkward integral)

[CanIExplore]

Homework Statement

What percentage of the Sun’s blackbody radiation spectrum falls into the visible light spectrum (400-700 nm). Where T=5000K
Hint: Integrate over frequencies

Homework Equations

B=2h\nu³c^-2 (e^h\nu/kT-1)^-1

Where \nu is the frequency of the light.

The Attempt at a Solution

Ok so the problem is very straightforward, I'm just having trouble evaluating the integral. I need to integrate the equation for the brightness (B) over the frequency \nu where the limits are given by the span of wavelengths in the visible part of the spectrum. The integral then just looks something like this:
\int B=\int \nu³(e^\nu-1)^-1 , where i excluded the constants.

I tried integration by parts but it didn't work. I also plugged it into mathematica and got a very weird, long answer that didn't make sense. When I asked my TA about it he told me I had to solve it numerically. What does it mean to solve an integral numerically? Am I just supposed to plug in the lower limit and then subtract that from what I get when I plug in the upper limit? I am thinking of the fundamental theorem of calculus here.

Thanks

 

[Cyosis]

This integral doesn't have a primitive in terms of elementary functions. To evaluate this integral you can use the Raleigh-Jeans approximation by looking it up in your book or by using the Taylor series of the exponent up to first order.

 

[phyzguy]

When you integrate a function numerically, you basically divide the area up into a bunch of narrow rectangles or trapezoids and add them all up. There are a lot of ways to do this, try reading this:

http://en.wikipedia.org/wiki/Numerical_integration

To accomplish this, you could write a program to do it, or use a canned program. Mathematica has a function NIntegrate which will do numerical integration and, given the function and the endpoints, will just return a number.

 

[CanIExplore]

Thanks guys