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This is directly from Tuckerman's Statistical Mechanics textbook, Chapter 12.6

The part that gets me confused starts from the 2nd line of eqn. (12.6.38) to the first line of eqn.(12.6.39), I can't follow how it's derived? It would be much appreciated if I can get some hints here!

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As physicists like doing, that's integration by parts, and neglecting the boundary term.

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    that's how I don't get it. Integration by parts has the form $\int uv' = uv - \int v du$, this case, we let $u = e^{-\beta \gamma}$ and $v' = x_k \frac{\partial}{\partial x_k} e^{-\beta \alpha}$, and then I can't simplify it into the form of first line in eqn. (12.6.39). Can you interpret more? And also I'm dubious about the double summation, might be a typo? – 2017-02-18
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    You have $v'=\frac{\partial}{\partial x_k} e^{-\beta\alpha}$ and $u=e^{-\beta\gamma}x_k$. The only thing that has a derivative before it is v, and the only thing after the IBP that has a derivative is u. Double summation seems fishy as both sums are done on k... – 2017-02-18
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    Especially since one of the sums disappears the next line over. – 2017-02-18
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    I'm still unable to obtain the desired form by your choice of $u$ and $v'$, but it does make sense to do it in your way. By doing IBP, I arrived upon the form $x_k e^{-\beta \gamma} e^{-\beta \alpha} + \beta \int x_k e^{-\beta \alpha} e^{-\beta \gamma} \frac{\partial \gamma}{\partial x_k} dx_k$, since $\frac{\partial}{\partial x_k} (e^{-\beta \gamma}) = - \beta e^{-\beta \gamma} \frac{\partial \gamma}{\partial x_k})$, which simplifies to $x_k e^{-\beta \gamma} e^{-\beta \alpha} - \int x_k e^{-\beta \alpha} \frac{\partial}{\partial x_k} \left[ e^{-\beta \gamma} \right]$ – 2017-02-18
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    then by keep evaluating the integral, it just gets messier, and I have trouble to get the first term cancelled...may I get a little more hint from you? Am I on the right track btw? – 2017-02-18
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    @GavyLittlewolf The first term is a boundary term. You have to evaluate it on the boundary of your integral. – 2017-02-18
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    The second term, at their first line, they didn't evaluate it. And also, it is not the term I wrote down. – 2017-02-18