Really simple question regarding integration by parts. I don't see why the integral of $x \cos(x) dx$ is what it is. Here's what I do, what am I doing wrong? I proceed by integration by parts, setting $f(x) = x$ and $g'(x) = \cos(x)$. So we have that $f'(x) = 1$ and $g(x) = \sin(x) + C$.
$\int x \cos(x) \ dx = x(\sin(x) + C_1) - \int\sin(x) + C_2 \ dx$.
Different constants since they are not necessarily the same, right?
$= x\sin(x) + xC_1 + \cos(x) - C_3 - xC_2$ $= x\sin(x) + \cos(x) + x(C_1 - C_2) + C_3$.
However, the solution shouldn't actually include that third term, so what's the deal?
When you take the (indefinite) integral you add a constant. When you derivate you do not add a constant: $$ f(x)+c \to (d/dx) \to f'(x) +0 $$ and look to it in the reverse and you will get the point.
Now $$ F(x) = f(x)g(x)+c\;\to\;F'(x)=(f(x)g(x)+c)'=f'(x)g(x)+f(x)g'(x)$$ but $$ F(x) = (f(x)+b)g(x)+c\;\to\;F'(x)=((f(x)+b)g(x)+c)'=f'(x)g(x)+f(x)g'(x)+bg'(x)$$ again, reverse the view going back and you'll get the point :i.e. the correct interpretation of indefinite summation by parts.