Q. Let $P_k$ be a point in $x y$ plane whose $x$ coordinate is $1+\frac{k}{n}(k=1,2,3, \ldots \ldots, n)$ on the curve $y=\ln x$. If $A$ is $(1,0)$, then $\underset{n \rightarrow \infty}{\text{Lim}} \frac{1}{n} \displaystyle\sum_{k=1}^n\left( AP _{ k }\right)^2$ equals
Integrals
Solution:
$L =\underset{n \rightarrow \infty}{\text{Lim}} \frac{1}{ n } \displaystyle\sum_{ k =1}^4\left(\left(\frac{ k }{ n }\right)^2+\ln ^2\left(1+\frac{ k }{ n }\right)\right)$
$=\int\limits_0^1\left(x^2+\ln ^2(1+x)\right) d x=\frac{1}{3}+2 \ln ^2 \frac{2}{e}$
