Question

$n\stackrel{Lt}{\to }\infty [\frac{n+1}{n^2+1^2}+\frac{n+2}{n^2+2^2}+\dots +\frac{n+n}{n^2+n^2}]=$

• A. $\frac{\pi }{4}+\frac{1}{2}\log 2$
• B. $\frac{\pi }{4}-\frac{1}{2}\log 2$
• C. $\frac{\pi }{2}+\frac{1}{2}\log 2$
• D. $\frac{\pi }{4}+\frac{1}{4}\log 2$

Answer 1

The correct option is A $\frac{\pi }{4}+\frac{1}{2}\log 2$

$n\stackrel{Lt}{\to }\infty {\sum }_{r=1}^n\left(\frac{n+r}{n^2+r^2}\right)=n\stackrel{Lt}{\to }\infty {\sum }_{r=1}^n\frac{1}{n}\left(\frac{1+\frac{r}{n}}{1+(\frac{r}{n}{)}^2}\right)$
$={\int }_0^1\left(\frac{1+x}{1+x^2}\right)dx={\int }_0^1\frac{1}{1+x^2}dx+\frac{1}{2}{\int }_0^1\frac{d{x}^2}{1+x^2}$
$=ta{n}^{-1}x{\int }_0^1+\frac{1}{2}log(1+x^2){\int }_0^1$
$=\frac{\pi }{4}+\frac{1}{2}log\left(2\right)$