How one integrate both sides of the inequality

Question

I have an inequality of the form:

$$u′(x)>-g′(x)$$ for $$x∈[a,c)$$

Here the interval is open.

My question is: How one integrate both sides of the inequality.

Answer 1

For $x_0, x \in [a,c)$: $$ \int\limits_{x_0}^x\! u'(\xi) \, d\xi > - \int\limits_{x_0}^x\! g'(\xi) \, d\xi \iff \\ u(x) - u(x_0) > -(g(x)-g(x_0)) $$

Answer 2

If you unknow $u'$ and $g'$ it not possible know the behavior of $u$ and $f$. Hence It not possible in one integrate both sides of the inquality know the behavior of the final expression.