Is it a good practice to write this integral in this form? I'm trying to compute the following integral:
$$\int e^{3x}\cos2x \;dx$$
Now I'm about to use the integration by parts. Suppose that I do not know what is the integral of $\displaystyle  \int \cos2x\; dx$. Is it a good practice to write it like this:
$$\int \stackrel{f}{e^{3x}}\cdot \stackrel{g'}{\cos2x} \;dx=e^{3x} \cdot\int \cos2x\; dx -\int 3e^{3x}\cdot  \left( \int \cos2x\; dx  \right) \; dx$$
And then compute the desired antiderivatives that I don't know later?
 A: Instead of using integration by parts, you may just assume that:
$$\int e^{3x}\cos(2x)\,dx = e^{3x}\left( A\sin(2x)+B\cos(2x)\right)+C $$
then find $A,B$ through differentiation. That is not much different from noticing that:
$$\int e^{3x}\cos(2x)\,dx = \text{Re}\int e^{(3+2i)x}\,dx = \text{Re}\left(\frac{1}{3+2i}\,e^{(3+2i)x}+C\right) $$
from which $A=\frac{2}{13},B=\frac{3}{13}$ follows.
A: That's sort of OK...but I like to think of the integral sign and the "dx" as being like open- and close-parentheses: they have to match. Furthermore, you're not allowed to have nested integrals with the same variable. [These are just my own rules for avoiding writing nonsense, not some axioms of mathematics! But if you write something that violates these rules, it probably won't make sense.]
To fix the "matching parens" problem, you'd need to change to 
$$
\int \stackrel{f}{e^{3x}}\cdot \stackrel{g'}{\cos2x} \;dx=e^{3x} \cdot\int \cos2x\; dx -\int 3e^{3x}\cdot  \left( \int \cos2x\; dx  \right) \; dx
$$
but then you'd have two nested integrals with the same variable of integration. You CAN write this as something like
$$
\int \stackrel{f}{e^{3x}}\cdot \stackrel{g'}{\cos2x} \;dx=e^{3x} \cdot\int \cos2x\; dx -\int 3e^{3x}\cdot  \left( \int^x \cos2t\; dt  \right) \; dx
$$
which says "compute the antiderivative with respect to $t$, and then evaluate at $t = x$". I've always disliked that notation, but it works for some people. 
The problem here is that we use the integral sign to denote two very different things: an antiderivative and a definite integral. These are, of course, very closely related -- by the Fundamental Theorem of Calculus -- but it's still a notational nightmare. If you use $\mathcal A_x (Q )$ to mean "the set of all  antidervatives of the function $x \mapsto Q$ (where $Q$ is some expression), then your formula would look like
$$
\mathcal A_x( \stackrel{f}{e^{3x}}\cdot \stackrel{g'}{\cos2x} ) = e^{3x} \cdot \mathcal A_x(\cos2x) -\mathcal A_x( 3e^{3x}\cdot  \mathcal A_x(\cos2x))
$$
which would at least be notationally consistent, although not as suggestive as the integral form. 
By the way, the "equality" here is one of sets -- the set of all antiderivatives on the left is equal to the set of functions you get on the right. Or, to put it differently, if you want $\mathcal A_x$ to mean "an antiderivative" rather than "the set of all antiderivatives", then you need to add "$+ C$" on one side or the other.  
