Explanation of the passage from $\int_{N'}^N dN/N$ to $\ln N-\ln N'$

Question

While going through my text I got stuck in the derivation given in the picture.

($\Omega$ is a constant)

I don't know how to get the second step from the first step, also I don't know why ln is applied in the second step.

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Translation (by an editor):

$$\int_{N'}^N dN/N = -\Omega \int_{t'}^t dt$$ $$\ln N-\ln N' = -\Omega (t-t')$$

Answer 1

The piece you seem to be missing is $\int{\frac{1}{x} dx}=\ln x +C$ Then $\int_M^N {\frac{1}{x} dx}=\ln N-\ln M$ You shouldn't have the same variable inside the integrand as in the limits of the integrand.