# Multiply and Integrate

Let $\Omega\subset\mathbb{R^2},$ $\phi\in L_2, t\in(0,T), T>0$ and $\gamma>0.$

$$|\xi|_{H^1(\Omega)}^2(t)\leq\gamma\int_{0}^T \left|\frac{\partial\phi}{\partial t}\right|_{L_2(\Omega)}^2dt$$

If we multiply with $(T-t)$ and then integrate from 0 to T for t, we have, $$\int_0^T(T-t)|\xi|_{H^1(\Omega)}^2(t)dt\leq\frac{1}{2}\gamma\int_{0}^T (T-t)^2\left|\frac{\partial\phi}{\partial t}\right|_{L_2(\Omega)}^2dt$$

Can anybody help me with the derivation for the right hand side of the last equation?