Does "partial integration" exist analogous to partial differentiation (in general)? I want to know whether "partial integration" exists analogous to partial differentiation in ordinary calculus for functions of several variables.
 A: First, please do note as pedja has mentioned in the comments above that the term "partial integration" is synonymous with "integration by parts".
What you're looking for might better be called something like "partial antiderivatives".
These things do show up frequently. Although I don't think they really have a definite name. Here are a couple of examples.
Suppose we wish to compute a double integral $\iint_R x^2y\,dA$ where $R$ is the rectangle $[0,1]\times [0,2]$. Then Fubini's theorem tells us that the double integral can be computed via iterated single integrals as follows:
$$ \iint_R x^2y\;dA = \int_0^1 \int_0^2 x^2y \;dy\;dx = \int_0^1 \left[ \frac{1}{2}x^2y^2 \right]_0^2 \;dx = \int_0^1 2x^2 \,dx = \left. \frac{2}{3}x^3 \right|_0^1 = \frac{2}{3} $$
Let's focus on the inner integral. I used the fact that $x^2y$ when holding $x$ constant and integrating with respect to $y$ is $\frac{1}{2}x^2y^2 + C(x)$. When plugging in endpoints the "$C(x)$" term disappears (just as the constant does not affect the value of a single definite integral). In some sense you could consider $\frac{1}{2}x^2y^2+C(x)$ the partial integral or partial antiderivative of $x^2y$ with respect to $y$. Although I don't think anyone uses such terminology. Also, notice that we have a whole function of $x$ instead of a constant because partial differentiation with respect to $y$ kills such things.
Another place this sort of thing shows up is when one finds potential functions for conservative vector fields. Let ${\bf F}(x,y)=\langle 3x^2+y, x+5 \rangle$.
We integrate the first component with respect to $x$ and the second component with respect to $y$ (these are your "partial" integrals): 
$\int 3x^2+y\;dx = x^3+xy+C_1(y)$ and $\int x+5\;dy = xy+5y+C_2(x)$. We want a function $f(x,y)$ such that $f(x,y)=x^3+xy+C_1(y)=xy+5y+C_2(x)$. The only way to reconcile these expressions is to let $C_1(y)=5y+$constant and $C_2(x)=x^3+$constant. So if $C$ is a constant, we have $f(x,y)=xy+x^3+5y+C$ where $\nabla f={\bf F}$.
