Question

Am nevoie de ajutor la acest exercițiu. Mulțumesc!

Answer 1

$\displaystyle I = \int_{\frac{1}{n+2}}^{\frac{1}{n}}\Big[\dfrac{1}{x}\Big]\, dx,\quad n\in \mathbb{N}^*\\ \\ \\\Big[\dfrac{1}{x}\Big] = \left\{\begin{array}{II}n+1,\quad x\in \Big(\dfrac{1}{n+2},\dfrac{1}{n+1}\Big) \\ \\n,\quad x\in \Big(\dfrac{1}{n+1},\dfrac{1}{n}\Big) \end{array}\right\\ \\\\ \Rightarrow I = \int_{\frac{1}{n+2}}^{\frac{1}{n+1}}(n+1)\, dx+ \int_{\frac{1}{n+1}}^{\frac{1}{n}}n\,dx =$

$=(n+1)x\Big|_{\frac{1}{n+2}}^{\frac{1}{n+1}}+nx\Big|_{\frac{1}{n+1}}^{\frac{1}{n}} =1-\dfrac{n+1}{n+2}+1-\dfrac{n}{n+1} = \\ \\ = 1-1+\dfrac{1}{n+2}+1-1+\dfrac{1}{n+1}= \\ \\ = \dfrac{n+1+n+2}{(n+1)(n+2)} = \dfrac{2n+3}{(n+1)(n+2)}\Rightarrow \boxed{A}-\text{Corect}$