Changing variable of integration

Question

So, I am a bit confused as to the subtle difference between something like $\int_0^5xdx$ and $\int_0^5x\frac {dx}{dt}dt$. Would these be computed differently? My thinking is that a small change in x per change in t multiplied by that change in t gives us the total change in x. In other words, is it correct to say $\frac {dx}{dt}dt = dx$? Would these two integrals have the same exact values? If not, why, and how would I know the how to correctly compute the two different integrals?

Answer 1

\begin{align*} \int_{0}^{5}x\dfrac{dx}{dt}dt=\int_{x(0)}^{x(5)}xdx\ne\int_{0}^{5}xdx \end{align*} in general, here $x'>0$ is assumed, and the left-sided is dealing with function $x=x(t)$ where the right-sided is dealing with function $x\rightarrow x$.