I am trying to solve a control systems problem and I am not sure if my argumentation is correct. $g(x_1)$ is a nonlinear function, meaning that it can be a hysteresis function with a positive offset or just $g(x_1) = |x_1|$.
$$ g(x_1) = \left\{ \begin{matrix} > 0 \; \forall\; x \in \; \mathbb{R}\setminus\{0\} \\ 0 \;for \;x_1 = 0\\ \end{matrix} \right. $$ Can I assume, that $ \int_{0}^{x_1}\eta\cdot g(\eta) d\eta = [ \,\eta\cdot G( \eta)] \,\Big|_0^{x_1} - \int_{0}^{x_1}G(\eta)d\eta = x_1\cdot G(x_1) - \int_{0}^{x_1}G(\eta)d\eta > 0 \; \forall\; x_1 \in \; \mathbb{R}\setminus\{0\}$ ?
As a integral is the area under the graph and the x-axis, the first expression $x_1\cdot G(x_1)$ should always be greater than $0$. If $x_1 <0 $ then $G(x_1) = \int_{0}^{x_1}g(\eta)d\eta = -\int_{x_1}^{0}g(\eta)d\eta\;$ should yield a negative sum and therefor $x_1\cdot G(x_1)$ is always greater than $0$ (negative times negative). If $x_1$ is positive, then the expression $x_1\cdot G(x_1)$ is also always positive.
I am not sure if this argumentation makes any sense.
As for the secound expression, $\int_{0}^{x_1}G(\eta)d\eta$ I am not really sure how to solve it or if I can even assume that is (in this case) $\int_{0}^{x_1}G(\eta)d\eta < 0\; \forall\; x_1 \in \; \mathbb{R}$.
Any input or tips would be helpful.
Best regards