- #1

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## Homework Statement

Hi everyone, I'd appreciate it if someone could help look through my working and check if it makes sense!

I have the following integral:

$$P(t) = \int_{-\infty}^{a} \int_{-\infty}^{-\infty} f_p(p) \ f_{x} (x - pt/m) \ dp \ dx$$

I want to find an expression for ##\frac{dP(t)}{dt}##, so I first "bring in" the integral over ##x##.

$$P(t) = \int_{-\infty}^{-\infty} f_p(p) \Big [ \int_{-\infty}^{a} f_x (x - pt/m) \ dx \Big ] \ dp$$

Making the substitution of ##u = x - pt/m##

$$\frac{d}{dt} \int_{-\infty}^{a} f_x (x - pt/m) \ dx = \frac{d}{dt}\int_{-\infty}^{a - pt/m} f_x (u) \ du = -\frac{p}{m} f_x(a - pt/m)$$

and finally,

$$\frac{dP}{dt} = -\frac{1}{m} \int_{-\infty}^{\infty} f_p (p) . p . f_x (a - pt/m) \ dp$$

Is this right? Many thanks in advance.