change of variables with limits

Question

If I have the following function:

$$ 1 = \int_0^{R_M}g_s(R)R^{-1}dR$$

where $x_M = R_M/\sigma_0$, and $x = R/\sigma_0$

how would I perform sub. of variables on the limits?

I know I would have something like:

$$ 1 = \int_0^{R_M}g_s(x)\frac{1}{\sigma_0x_m} \sigma_0 dx$$

$$ 1 = \int_0^{R_M}g_s(x)\frac{1}{x_m} dx$$

What would the new limits be? Would it be $x_M$ or $x_M\sigma_0$?

Answer 1

1. The limit is $x_M$
2. The function in the integral should be $g_s(\sigma_0 x)$