(Solved): 3. Integrate \( \int 2 h c^{2} / \lambda^{5}(\exp (h c / \lambda k T)-1) d \lambda \) from 0 to \( ...

3. Integrate \( \int 2 h c^{2} / \lambda^{5}(\exp (h c / \lambda k T)-1) d \lambda \) from 0 to \( \infty \), to show \( B(T)=\left[2 \pi^{4} k^{4} / 15 c^{2} h^{3}\right] T^{4} \). [HINT: Let \( x=h c / \lambda k T \) and don't forget about \( d \lambda \). Also, let \( \int x^{3} /\left(e^{x}-1\right)=\pi^{4} / 15 \). When you are done, put into words what you just did, everything you think your result produced and what it means in general terms (i.e., what did this accomplish).