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$$I = \int\limits_{t_a}^{t_b}{\left(\frac{d{x}}{d{t}}\right)^2 \mathbb dt}$$

$x_a = x(t_a)$ 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 $I$?

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Take some examples of functions $x(t)$ and see what you get. Say, $x(t) = t^3$ or $x(t) = \sin(t)$. – GEdgar Jul 9 '11 at 0:25
Any additional information available? – André Nicolas Jul 9 '11 at 2:29
$x$ is a straight line in 3D. – Pratik Deoghare Jul 9 '11 at 11:12

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.

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Integrating by parts could simplify things, (not only) when $\ddot{x}$ is a constant: $$ I = \int \dot{x}^2 dt = \int \dot{x} dx = \dot{x}x - \int x\ddot{x} dt $$

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I don't believe this is equivalent since we have $\left(\frac{dx}{dt}\right)^2$, which means we really have two of them, thus one integral is not enough to sum all of the $dx$'s with respect to $dt$'s... – jm324354 Oct 20 '14 at 23:00

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