Problem from Eisberg: Blackbody Radiation

[raghav]

Homework Statement

In case of Cavity Radiation(Blackbody radiation) let $$\rho_{T}(\nu)d\nu$$ denote the energy density of radiation having frequency in the interval $$\nu$$ and $$\nu + d\nu$$.
Then we need to show that $$\frac{\int_{0}^{\lambda_{max}} \rho_{T}(\nu)d\nu}{\int_{0}^{\infty} \rho_{T}(\nu)d\nu} \approx \frac{1}{4}$$
where $$\lambda_{max}$$ is the wavelength at which the radiation is most intense.

Homework Equations

The obvious rela\evant equation is
$$\rho_{T}(\nu)d\nu = \frac{8\pi h\nu^{3}}{c^{3}}\cdot \frac{d\nu}{e^{\frac{h\nu}{kT}}-1}$$ . But the problem is the integration part. Can some one please help me with that?

The Attempt at a Solution

 

[BerryBoy]

OK, so:

$$\int \nu ^3 \cdot (e^{\frac{h\nu}{k_b T}}-1)^{-1} \cdot \delta \nu$$

Is the equation you need to solve, so you can use:

$$\int A \delta B = AB - \int B \delta A$$

(from the differentiation product rule). There's a start. Let me know if it helped.

Sam

 

[BerryBoy]

Oops, and you'll probably need this standard integral:

$$\int_0^{\infty} \frac{x^3}{e^x -1}\delta x = \frac{\pi^4}{15}$$

Sam

 

[raghav]

Oops, and you'll probably need this standard integral:

$$\int_0^{\infty} \frac{x^3}{e^x -1}\delta x = \frac{\pi^4}{15}$$

Sam

Yes i could evaluate the denominator using the standard integral u have mentioned, however the Numerator is creating trouble. I have also tried a parts argument, doesn't seem to be taking me any where
Some1 please help asap

 

[BerryBoy]

I can't do this without some assumptions, have you stated the WHOLE question from the start?

Sam

 

[raghav]

Yes I have indeed given all the necessary details the problem demands. Essentially my doubt boils down to evaluating the integral
$$\int_{0}^{a} \frac{t^{3}}{e^{t}-1}dt$$ ; $$a$$ being some constant.