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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