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Number Theory

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Easy

Let $n > 1$. What is $\int_{1}^{n}\frac{dx}{x}$?

NUMTH-AYS5XN

Let $n \in \mathbb{N} \backslash \{1\}$. Then, by the Fundamental Theorem of Arithmetic, $n = p_{1}^{\alpha_1}\cdots p_{s}^{\alpha_s}$ for some $s \in \mathbb{N}$, where $p_1, ..., p_s$ are distinct prime numbers and $\alpha_1, ... \alpha_s \in \mathbb{N} \cup \{ 0 \}$. What is $\int_{1}^{n}\frac{dx}{x}$?

A

$\sum_{j = 1}^{s} \alpha_{j} \log p_j$

B

$\prod_{j = 1}^{s} \alpha_j \log p_j$

C

$\left(\sum_{j = 1}^{s} \alpha_ j\right)\left(\sum_{j = 1}^{s} \log p_j\right)$

D

$-\cfrac{1}{\left(p_{1}^{\alpha_1}\cdots p_{s}^{\alpha_s}\right)^2}$