# Limits and Integrals!

Calculus Level 4

$\displaystyle \lim_{n\rightarrow\infty}\left(\dfrac{2n}{n^{2}+1}+\dfrac{2n}{n^{2}+4}+\dfrac{2n}{n^{2}+9}+\cdots+\dfrac{2n}{n^{2}+n^{2}}\right)$

Let $$A$$ denote the value of the limit above. And if the value of, $\displaystyle I=\int_{0}^{A}\dfrac{x}{\sin(x)+\cos(x)}dx$

is of the form, $\displaystyle \dfrac{\pi}{a\sqrt{b}}\left(\ln(\sqrt{c}+d)-\ln(\sqrt{e}-f)\right)$

Find the value of $$a+b+c+d+e+f$$.

Details and Assumptions:

1) $$a,b,c,d,e,f$$ are integers . They need not to be distinct

2) $$b,c,e$$ are not having any factor which is a perfect square.

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