Answer
Prove:
$\mathop \smallint \limits_c^d f\left( {{{\bf{r}}_1}\left( t \right)} \right)||{{\bf{r}}_1}'\left( t \right)||{\rm{d}}t = \mathop \smallint \limits_a^b f\left( {{\bf{r}}\left( t \right)} \right)||{\bf{r}}'\left( t \right)||{\rm{d}}t$
Work Step by Step
We have the re-parametrization ${{\bf{r}}_1}\left( t \right) = {\bf{r}}\left( {\varphi \left( t \right)} \right)$, where $\varphi \left( t \right)$ is an increasing function.
Taking the derivative of ${{\bf{r}}_1}\left( t \right)$, we get
${{\bf{r}}_1}'\left( t \right) = {\bf{r}}'\left( {\varphi \left( t \right)} \right)\varphi '\left( t \right)$
$||{{\bf{r}}_1}'\left( t \right)|| = ||{\bf{r}}'\left( {\varphi \left( t \right)} \right)||||\varphi '\left( t \right)||$
Since $\varphi \left( t \right)$ is an increasing function, we can just write
$||{{\bf{r}}_1}'\left( t \right)|| = ||{\bf{r}}'\left( {\varphi \left( t \right)} \right)||\varphi '\left( t \right)$
Substituting this in the scalar line integral with parametrization ${{\bf{r}}_1}\left( t \right)$, we obtain
$\mathop \smallint \limits_c^d f\left( {{{\bf{r}}_1}\left( t \right)} \right)||{{\bf{r}}_1}'\left( t \right)||{\rm{d}}t = \mathop \smallint \limits_c^d f\left( {{\bf{r}}\left( {\varphi \left( t \right)} \right)} \right)||{\bf{r}}'\left( {\varphi \left( t \right)} \right)||\varphi '\left( t \right){\rm{d}}t$
But $d\left( {\varphi \left( t \right)} \right) = \varphi '\left( t \right){\rm{d}}t$. So,
$\mathop \smallint \limits_c^d f\left( {{{\bf{r}}_1}\left( t \right)} \right)||{{\bf{r}}_1}'\left( t \right)||{\rm{d}}t = \mathop \smallint \limits_a^b f\left( {{\bf{r}}\left( {\varphi \left( t \right)} \right)} \right)||{\bf{r}}'\left( {\varphi \left( t \right)} \right)||{\rm{d}}\left( {\varphi \left( t \right)} \right)$
where $a = \varphi \left( c \right)$ and $b = \varphi \left( d \right)$.
By replacing the variable $\varphi \left( t \right)$ on the right-hand side by a dummy variable $t$, we obtain
$\mathop \smallint \limits_c^d f\left( {{{\bf{r}}_1}\left( t \right)} \right)||{{\bf{r}}_1}'\left( t \right)||{\rm{d}}t = \mathop \smallint \limits_a^b f\left( {{\bf{r}}\left( t \right)} \right)||{\bf{r}}'\left( t \right)||{\rm{d}}t$
which is to be proved.
