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For part a) you should check that the symmetry of $P(x)$ with respect to $x$ gives
$$\int_{-\infty}^{\infty}Ne^{-|x|/a} dx=2\int_{0}^{\infty}Ne^{-x/a}dx$$
and then do a change of variable $\displaystyle u=\frac{x}{a}$
For part b) note that by definition $\displaystyle\langle f(x) \rangle=\int_{-\infty}^{\infty}f(x)Ne^{-|x|/a}dx$ and that both $x$, $x^2$ ...
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There are closed formulae for the heat capacities of a Gibbsian substance which is desribed by two equations of the form $T=f(p,V), S=g(p,V) $. They are $C_V=\frac{fg_p}{f_p}$ and $C_p=\frac{fg_V}{f_V}$ where I am using subscripts to indicate partial derivatives. The details can be found in the arXiv paper 1102.1540 (of which I am a co-author) which also ...
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