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I was reading a paper and couldn't understand the following transition. Could someone tell me where the term of $p^k (\frac{1}{2} − c′)$ comes from in the following transition?

Def: Cumulative function of interest

$$ F(x) = \begin{cases} 0 & \text{for } x < -c' \\ p^k & \text{for } x \in [-c', 0) \\ (p + (1 - p) x)^k & \text{for } x \in [0, 1] \\ 1, & \text{for } x > 1 \end{cases} $$

Integral of interest

$$ \int^{\infty}_{-\infty} x d F(x) = p^k (\frac{1}{2} - c') + \int^{1}_{0} x d F(x) = \cdots $$

My attempts

First, I expanded the integral based on the sections as follows; $$ \int^{\infty}_{-\infty} x d F(x) = \int^{-c'}_{- \infty} x dF(x) + \int^{0}_{-c'} x dF(x) + \int^{1}_{0} x dF(x) + \int^{\infty}_{1} x dF(x) $$ Then, i was not sure how to compute each term... could someone tell me how to compute the integral over a CDF?

Reference

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