Does integration by parts works for partial derivatives? Can we write $$\int_a^b \frac{\partial f(x,y)}{\partial x}g(x,y) dx = f(x,y)g(x,y)|_a^b - \int f(x,y)\frac{\partial g(x,y)}{\partial x}dx$$
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2You forgot $dx$ at the end. It works as you have typed, just as long as you look at $x\mapsto f(x,y)$ and $x\mapsto g(x,y)$ as one-variable functions. – Git Gud Dec 30 '13 at 15:32
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Yes it does, for fixed $y$. When you integrate with respect to $x$ we hold $y$ fixed, therefore it is treated as a constant. In other words,
$$\int_a^b x^2 e^{kx} \, dx \quad \text{is equivalent to} \quad \int_a^b x^2 e^{xy} \, dx.$$
There are double/triple integral identities which are known as multivariable integration by parts (Green identities).
Mark Fantini
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