Solution of integral equation

Question

If $x$ is a real-valued, differentiable function of $t$, what is, and how do I find the solution of

$$\int_a^b x(t) \frac{dx(t)}{dt} dt$$

Answer 1

Fundamental theorem of calculus:

$${1\over 2}x(t)^2\bigg|_a^b={1\over 2}\left(x(b)^2-x(a)^2\right).$$

It comes from the fact that

$$dx={dx\over dt}\,dt={dx(t)\over dt}\,dt$$