We know that $$\int_\gamma V \cdot dr = \int_a^b V(r(t)) \cdot r'(t) dt$$ with $V$being our vector field and $r$being the parametric equation for the curve $\gamma$.
Let now $\hat{r} = r \circ \phi$ for some appropiate function $\phi$. How do we show that $$\int_{a}^b V(r(t)) \cdot r'(t) dt = \int_{c}^d V(\hat{r}(t)) \cdot \hat{r}'(t) dt?$$ I insert the definition of $\hat{r}$ in the RHS and work around, but then get stuck at $$\int_c^d V(r(\phi(t)) \cdot r'(\phi(t))\phi'(t)$$ ($[c,d]$ being the domain for $r$ hat). Using $u$-substitution ends with me getting away from what i want to show, but that still seems the way to go?