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$$\frac{i \hbar}{2m} \int_\text{region} \left((\nabla^2\phi^*) \phi - \phi^* (\nabla^2 \phi) \, \,\right)\mathrm{d}x$$

How would I be able to solve this integral using integration by parts or other methods?

($\phi$ is a wave function and $\phi^*$ is used as conventionally.)

This question can be divided into two:

1) When region is $\mathbb{R}^n$

2) When region is some finite shape

3) Should this equation be corrected to $$\frac{i \hbar}{2m} \int_\text{region} \left((\nabla^2\phi^*) \phi - \phi^* (\nabla^2 \phi) \, \,\right)\mathrm{d}x \ \mathrm{d}y \ \mathrm{d}z$$?

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Did you try to use one of Green's identities? The "on manifolds" part of the article tells you the answer to 2). –  filmor Oct 4 '12 at 8:12
    
Regarding 3): It is very common in mathematics to denote elements of a vector space without any decoration, so $\mathrm dx$ is the same as $\mathrm dx_1\,\mathrm dx_2\,\mathrm dx_3$. Another interpretation would be that $\mathrm dx$ is the Lebesgue measure on $\mathbb{R}^3$. If you only use one integral sign, only write down one measure. –  filmor Oct 4 '12 at 9:20
    
This has nothing to do with contour-integration. –  joriki Oct 4 '12 at 9:20
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