Is there such a thing as partial integration?

Your partial integral is roughly the same as your regular integral, with a caveat. If you have, say, $$\int \frac{d}{dx} f(x) dx$$ When you integrate this you end up with $f(x) + C$ - since this is the antiderivative of $f'(x)$, the $C$ shows up because integration only knows 'so much' - the derivative of $C$ is zero, so we don't know whether or not it's actually in $f(x)$. Similarly, when we take an integral over one variable, we get $$\int \frac{\partial}{\partial x} f(x,y) dx$$ The partial 'knocks out' any functions of $y$ in $f(x,y)$; for example, if $f(x,y)=xy+y^2$, then the partial will send $y^2$ to zero. So as before, when we integrate solely with respect to $x$ of a multivariable function, we get $$\int \frac{\partial}{\partial x} f(x,y) dx = f(x,y)+C(y)$$ Where $C(y)$ denotes any function of $y$. There's no way to get an integral that will 'invert' the partial operator while still knowing about $y$ - that information is lost when we took the partial in the first place.

Such integration is indeed used for certain purposes, for example, when you are looking for the antiderivative(potential) of the vector field: $\vec F(x,y)=(2xy,x^2)$. Then you need to find a scalar function $V(x,y)$ such that $\frac {\partial V}{\partial x}=2xy$ and $\frac {\partial V}{\partial y}=x^2$. Using indefinite integration, we can find $V=x^2y+C$ for constant $C$.

However, this idea is contained in the usual single variable indefinite integration: we just treat the integration of $f(x,y)$ w.r.t $x$ as the integration of the single variable function $g_y(x):=f(x,y)$ for any fixed $y$. Therefore we don't need to define a partial integration. Although the partial derivative has a definition also in this manner, but that concept is important because of its connection with total derivative.