Why don't we write the second constant While doing integration by parts we usually do not write the constant derived from the second part of the formula and instead we assign general constant for the whole integration.What is the reason for that?
 A: Let's consider what this constant means. When you write, for example, $\int 2x dx = x^2 + C$, this essentially means that $2x$ has a lot of antiderivatives: $x^2$ is one, $x^2 + 1$ is another, $x^2 - 3.272$ is a third, and so on. You can put any real number where the $C$ is, and you'll get a proper antiderivative.
What would happen if you have two constants? Say, we have $x^2 + C_1 + C_2$. This means we can replace $C_1$ with any real number, and $C_2$ with any real number. Like, we can choose $C_1=3$ and $C_2 = -3.15$, and get $x^2 + 3 - 3.15$. But anything we can achieve by adding two real numbers, we can also achieve by adding only one - their sum (in our example $x^2 - 0.15$). So having two constants doesn't give us anything more than just having one constant.
A: I think the question is being misinterpreted. The formula for integration by parts is $$\int f(x) g(x) \, dx=f(x) \int g(x) \, dx-\int\left(f'(x) \int g(x) \, dx\right) \, dx$$ Suppose we use anti-derivative $G(x)$ of $g(x) $ then the above formula can be written as $$\int f(x) g(x) \, dx=f(x) G(x) - \int f'(x) G(x) \, dx$$ If instead of $G(x) $ we use an anti-derivative $G(x) + C$ then we get $$\int f(x) g(x) \, dx=f(x) G(x) + Cf(x) - \int f'(x) G(x) \, dx-C\int f'(x) \, dx$$ The effect of last term is cancelled by the second term $Cf(x) $ and at most a constant remains as expected and thus we don't really need to add a constant explicitly for the anti-derivative of $g(x) $. 
