# Where does the identity $p^{-ns}=s\int_{p^n}^\infty x^{-s-1}dx$ come from? [closed]

In this paper, equation 45 (page 11) gives the identity

$$p^{-ns}=s\int_{p^n}^\infty x^{-s-1}dx$$

Can someone tell me where this comes from, and how it is derived?

• For $s\ne0$ the anti-derivative of $x^{-s-1}$ is $-\frac{x^{-s}}s$. – Shubham Johri Jan 7 at 15:41
• Thanks @Shubham Johri. Do you want to turn this into an answer so I can tick it? – Richard Burke-Ward Jan 7 at 15:43

This statement holds only for $$s>0$$.
For $$s\ne0$$ the anti-derivative of $$x^{-s-1}$$ is $$-\frac{x^{-s}}s$$. Applying the limits,$$\frac s{-s}[x^{-s}]_{p^n}^\infty=p^{-ns}$$since $$\infty^{-s}=1/\infty^s=0$$ for positive $$s$$.