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. raw materials 1. city in the sky: a story of hollywood s deadliest night 2. chapter 7: the storm 3. chapter 8: the storm: aftermath 4. carman rust .Q:

Solve an integration

Need to solve following integral.
$$\int_{2}^{\infty} \frac{x^2}{e^{x}-1} dx$$

A:

This is a complete solution (I’m not sure that you need to explain step by step what you are doing, because you are looking for a general solution, but in any case I will write it).
I find it easier to start with
$$\int_{2}^{\infty} \frac{x^2}{e^{x}-1} dx = \sum_{k=1}^{\infty} \frac{1}{k} \int_{2^k}^{2^{k+1}} x^2 e^{ -x} dx$$
Now, using the substitution $u=2^{x}$ and $du = 2^x \: dx$ we get
$$\int_{2^1}^{2^2} x^2 e^{ -x} dx = \int_{2}^{\infty} u^{ -2} u^2 e^{ -u} du = \frac{1}{2} \int_{2}^{\infty} (1+u)^2 e^{ -u} du$$
Now we use the integration by parts to get
$$\frac{1}{2} \left(1-e^{ -u}u\right) = \left(1-\frac{1}{u}+\frac{1}{2} u^{ -2} \right) e^{ -u} u \implies \frac{1}{2} e^{ -u} u = 1 – \frac{1}{2} u^{ -1} + \frac{1}{2} e^{ -u}$$
We finally have
$$\frac{1}{2} \int_{2}^{\infty} e^{ -u} u dx = 1 – \frac{1}{2} \int_{2}^{\infty} \left(\frac{1}{u} + e^{ -u}\right) du$$
Now we simplify further using the asym