Integration by parts: $x f(x) dx$ Suppose I want to integrate:
$$
\int_a^b x f(x) dx
$$
Then if I know a function $F(x)$ such that $F'(x) = f(x)$, integration by parts tells me:
$$
\int_a^b x f(x) dx = \left[ x F(x) \right]_a^b - \int_a^b F(x) dx
$$
The question I am asking is: $F(x)$ is not unique since I can add any constant and still find $F'(x) = f(x)$. So how does one evaluate the integral on the right hand side if $F(x)$ is not uniquely defined?
 A: Despite the fact that antiderivative is not unambiguously defined, the right side will always give the same answer
$$\left[ x (F(x)+c) \right]_a^b - \int\limits_a^b (F(x)+c) dx=\\
=\left[ x F(x) \right]_a^b+\left[ x c \right]_a^b-\int\limits_a^b F(x) dx-\int\limits_a^b c dx$$
now desired follows from
$$\left[ x c \right]_a^b=\int\limits_a^b c dx$$
A: Suppose that $F$ is any function such that
$$
F'(x) = f(x) 
$$
Using integration by parts, we get
$$
\int_a^b x f(x) dx = \left[ x F(x) \right]_a^b - \int_a^b F(x) dx \tag{1}
$$
We define $G(x) = F(x) + c$.
Then it follows that
$$
G'(x) = F'(x) = f(x)
$$
Using (1), we can write
$$
\int_a^b x f(x) dx = \left[ x G(x) \right]_a^b - \int_a^b G(x) dx \tag{2}
$$
Since $G(x) = F(x) + c$, we can expand (2) as
$$
\int_a^b \ x f(x) dx = \left[ x [F(x) + c] \right]_a^b - \int_a^b \ [F(x) + c] dx  
$$
That is,
$$
\int_a^b \ x f(x) dx = \left[ x F(x) \right]_a^b +
\left[ x c \right]_a^b - \int_a^b \ F(x) dx   - \int_a^b \ c dx \tag{3}
$$
Simplifying (3), we get
$$
\int_a^b \ x f(x) dx = \left[ x F(x) \right]_a^b +
c (b - a) - \int_a^b \ F(x) dx   - c (b - a) dx 
$$
Thus, we get
$$
\int_a^b \ x f(x) dx =  \left[ x F(x) \right]_a^b - \int_a^b \ F(x) dx 
$$
