Very Simple Question, please help: Integral of a derivative squared

gilgtc

Hello,

I am trying to figure out how to integrate this, I know it must be simple but I am not sure how to do it.

[tex]\int(\frac{dx}{dt})^{2}dt[/tex]


Thanks a lot!

dirk_mec1

I think partial integration can work.

Nick R

You just need x defined in terms of t

x = x(t)

then you can differentiate with respect to t,

then you square dx/dt

then you integrate that across t from t1 to t2

right?

dirk_mec1

Wait a second is x(t) explicitly known?

gilgtc

hi, thanks for your answers. x(t) is not known that is why I am not sure how to do it. Otherwise what Nick mentioned would be easily applicable.

Any other ideas? What do you mean by partial integration dirk_mec1?

Thanks!

HallsofIvy

He means "integration by parts". Do you have any reason to think that there is any simple answer to this question? I can see no reason to assume that
[tex]\right(\frac{dx}{dt}\)^2[/tex]
even has an elementary anti-derivative.

dirk_mec1

I dont think you can just integrate [tex] \int (f'(x))^2 \mbox{d}x[/tex], right? The integration by parts(thanks hallsofIvy ) however gives:

[tex] x (x'(t))^2 - \int x \cdot 2x' \cdot x''\ \mbox{d}t [/tex]

gilgtc

I thought that there was an easy equivalence like:

[tex]\int(dx/dt)dt = x[/tex]

I guess not! thanks for your help in any case.