analytical approximation of unsolvable integral

[noou]

Hello everybody,

I need to "solve" an unsolvable integral which looks simple but is not:

there are three variables:
\(\displaystyle t, x(t), v(t)=\frac{dx}{dt}\)
and a constant
\(\displaystyle a\geq1\)
moreover I don't know anything about \(\displaystyle x(t)\)

the indefinite integral is
\(\displaystyle \int{x(t)^a v(t)^2 dt }\)

Any help is greatly appreciated! :wink:

Best,
Stefano

 

[DrMike]

This looks like a physics problem.

Do you have any other information?

 

[noou]

This looks like a physics problem.

indeed it is: it comes from a nonlinear impact model where a lumped mass impact a rigid wall.
the impact is modeled as a mass-nonlinear spring-damper system

Do you have any other information?

the initial equation is:
\(\displaystyle ma(t) = -kx(t)^\alpha - \lambda x(t)^\alpha v(t)\)
\(\displaystyle x\) is the displacement
\(\displaystyle v\) the velocity
\(\displaystyle t\) the time
\(\displaystyle \alpha\) is a constant of nonlinearity
\(\displaystyle k\) is the elastic constant
\(\displaystyle \lambda\) is the viscoelastic constant

for simplicity's sake, in my first post I assumed \(\displaystyle k\) and \(\displaystyle \lambda\) equal to 1
and I renamed \(\displaystyle \alpha\) to \(\displaystyle a\).

Thanks!