Computing definite integrals (with force)

Question

Say we wished to compute the change in momentum of a body, which undergoes a time-variable force. Then: $p(t)|^{t_f}_{t_i}=\int_{t_i}^{t_f}F(t)dt$.

Could you please explain why $\int_{t_i}^{t_f}F(t)dt = F^2(t)/2 + C$ is mathematically incorrect?

Answer 1

You are getting confused with $$\int t\ dt=\frac12t^2+C$$

But $F(t)$ can be any function of $t$. So that is not how you do the integral. In other words, you would be right if we were integrating with respect to $F$, but the integral is with respect to $t$

You can also see this by taking the time derivative of your proposed solution. $$\frac{d}{dt}\left(\frac12F^2(t)\right)=F\frac{dF}{dt}$$ Which is not $F$ unless $\frac{dF}{dt}=1$ , which means that $F(t)=t$, consistent with what is above.

I'm also voting to move this to Mathematics SE, since it is a math question.