integration by parts partial derivatives

Question

I have a question about integration by part and application in case with partial derivative. Is the following true ? $$ \int^{+\infty}_{-\infty} f(x,y)g(x,y)dx=f(x,y)G(x,y)-\int^{+\infty}_{-\infty}\frac{\partial f(x,y)}{\partial x} G(x,y)dx $$ where $$G(x,y)=\int^{+\infty}_{-\infty}g(x,y)dx$$ I am not sure that I can define $G$ that way. Thank you