Suppose that I have an arbitrary differentiable function $G$ which depends on $s$: $G(s)$.
Now suppose that I want to do the following integral: $$\int ds \;\; s \frac{\partial G(s)}{\partial s}.$$
However, I do not know the exact form of $G(s)$, so I cannot do the integral; I can only "expand" it. One method to do this is integration by parts. I think that integration allows me to rewrite the integral as $$\boxed{\int ds \;\; s \frac{\partial G(s)}{\partial s} = sG(s) - \int ds \, G(s)},$$
but I would like to verify this to make sure I am not making any mistakes. Is there any way that I can use Mathematica to verify that for the boxed equation, the left-hand side indeed equals the right-hand side?
I have tried this:
Integrate[s*D[G[s], s], s] == s*G[s] - Integrate[G[s], s]
but Mathematica (I am running version 8) does not know how to automatically evaluate this.
True
. So I'd say this is a duplicate of the linked question. $\endgroup$