TITLE: Does integration by parts work for partial derivatives?
QUESTION [5 upvotes]: Does integration by parts works for partial derivatives? Can we write $$\int_a^b \frac{\partial f(x,y)}{\partial x}g(x,y) dx = f(x,y)g(x,y)|_a^b - \int f(x,y)\frac{\partial g(x,y)}{\partial x}dx$$

REPLY [4 votes]: 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).