In a certain textbook, I see the Cumulative Distribution Function (CDF) of a continuous random variable X defined as $$\int_{-\infty}^{x'} dp(x)$$ where p(x) is the Probability Density Function of X.

Usually, I see the CDF defined thus: $$\int_{-\infty}^{x'} p(x)dx$$ I have only ever seen and solved integrals with respect to a variable, not with respect to a function of a variable, so I don't know if those two are equivalent. Are they? Or is the first expression wrong?