Double Integral with abstract functions of 2 variables I am required to prove something, and so far I have come to set up an integral 

$$\int_0^l{\int_0^T{u(x,t)\, \frac{d}{dt}u(x,t) dt }dx}.$$  

I was just wondering how to think about these integrals and approach them because they seem to be very confusing when they are not given functions. Thanks in advance! (The outer integral is an integral over the length, not from 0 to 1)

In the inner integral we regard the function $u(x,t)\frac{\partial}{\partial t}u(x,t)$ as a function in $t$. Note that the primitive with respect to $t$ is $\frac{1}{2}u(x,t)^2$. So you get
$$
\int_{0}^{1}\frac{1}{2}u(x,T)^2 -\frac{1}{2}u(x,0)^2dx.
$$
Now it's an integral in one variable, namely $x$ at which point you need more information about $u(x,t)$.
