STD 12 - 7.2 definite integral — Mathematics STD 12 Science — Question

$= \mathop {{\rm{lim}}}\limits_{n \to \infty } \,\left[ {\frac{1}{n} + \frac{1}{{n + 1}} + \frac{1}{{n + 2}} + .... + \frac{1}{{n + n}}} \right]$

$ = \frac{1}{n}\mathop {{\rm{lim}}}\limits_{n \to \infty } \left[ {1 + \frac{1}{{1 + \frac{1}{n}}} + \frac{1}{{1 + \frac{2}{n}}} + .... + \frac{1}{{1 + \frac{n}{n}}}} \right]$

$ = \frac{1}{n}\mathop {{\rm{lim}}}\limits_{n \to \infty } \sum\limits_{r = 0}^n {\left[ {\frac{1}{{1 + \frac{r}{n}}}} \right]} $ $ = \int_0^1 {\frac{1}{{1 + x}}\,\,dx} $

$ = [{\log _e}(1 + x)]_0^1 = {\log _e}2 - {\log _e}1 $

$= {\log _e}2$.