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.



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