$I = \int\limits_{t_a}^{t_b}{\left(\frac{d{x}}{d{t}}\right)^2 \mathbb dt}$
$x_a = x(t_a)$ and $x_b = x(t_b)$
I haven't integrated anything like this since a long time. Lost my powers of integration.
How do I calculate $I$?
$I = \int\limits_{t_a}^{t_b}{\left(\frac{d{x}}{d{t}}\right)^2 \mathbb dt}$
$x_a = x(t_a)$ and $x_b = x(t_b)$
I haven't integrated anything like this since a long time. Lost my powers of integration.
How do I calculate $I$?
Without knowing what function $x$ is of $t$, you can't do much. You could integrate by parts, letting $u = dx/dt$ and $dv = (dx/dt)\,dt$, and then you get $du = (d^2 x /dt^2) \,dt$ and $v = x$. But that doens't necessarily shed any more light than does the expression you've got already.
Integrating by parts could simplify things, (not only) when $\ddot{x}$ is a constant: $ I = \int \dot{x}^2 \, dt = \int \dot{x} \, dx = \dot{x}x - \int x\ddot{x} \, dt $