In kinetic energy equation in wiki.
I have difficulty problem how endpoint in integral change from t to v.
This doesn't look like it is using substitution method.
$\mathcal{E}_{k}=\int_{0}^{t}\boldsymbol{F}\ dx=\int_{0}^{t}\boldsymbol{v}\cdot d(m\boldsymbol{v})=\int_{0}^{v}d(\frac{mv^2}{2})=\frac{mv^2}{2}$
I'm not sure how to relate if you cancel differential dx as in above,
you can change the interval endpoint [0,t] to [0,v].
Let me write the offending calculation with correct change of bounds. Suppose $F=ma$ where $a = dv/dt$ then suppose $E_k = \int_0^x F dx$. We calculate, $$ E_k = \int_{x_o}^{x_f} F dx =\int_{t_o}^{t_f} \frac{dx}{dt}dt =\int_{t_o}^{t_f} mv\frac{dv}{dt} dt = \int_{v_0}^{v_f} mvdv = \frac{1}{2}mv_f^2-\frac{1}{2}mv_o^2$$ where I have used $t_o$ for the initial time and $t_f$ for the final time etc.