Change of variables in integral: the limits of $y = \int_{\sigma_0}^{R_M} \frac{dR}{R} $

Question

I have this equation:

$$y = \int_{\sigma_0}^{R_M} \frac{dR}{R} $$

where $x = R/\sigma_0$

If I want to do a change of variables, I would have:

$$y = \int \frac{\sigma_0}{x\sigma_0} $$

$$y = \int \frac{1}{x} $$

My question is: What do the limits become?

Answer 1

$R\; $ becomes $\; x\sigma_0$.

$dR\; $ becomes $\; \sigma_0dx$.

$\sigma_0\; $ changes to $\; \frac{\sigma_0}{\sigma_0}$.

and

$R_M$ will become $\frac{R_M}{\sigma_0}$.

the integral is then

$$\int_1^{\frac{R_M}{\sigma_0}}\frac{\sigma_0 dx}{x\sigma_0}=$$ $$\int_1^{\frac{R_M}{\sigma_0}}\frac{dx}{x}.$$

Answer 2

$R=\sigma _0$ implies that $ x= \frac {\sigma_0}{\sigma}$

$R=R_M $implies that $x=\frac {R_M}{\sigma}$

Thus the new bounds are the old ones divided by $\sigma$