Limits and Integrals!

Calculus Level 4

limn(2nn2+1+2nn2+4+2nn2+9++2nn2+n2)\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 AA denote the value of the limit above. And if the value of, I=0Axsin(x)+cos(x)dx\displaystyle I=\int_{0}^{A}\dfrac{x}{\sin(x)+\cos(x)}dx

is of the form, πab(ln(c+d)ln(ef))\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+fa+b+c+d+e+f.

Details and Assumptions:

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

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

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