For $a\ge 2$ , if  the value of the definite integral  $\underset{0}{\overset{\infty }{\int }}\frac{dx}{a^2+{(x-\left(\frac{1}{x}\right))}^2}$ equals $\frac{\pi }{5050}$  and the value of  a is four digit number PQRS then P + Q + R-S is equal to 

$$\begin{gathered} I=\underset{0}{\overset{\infty }{\int }}\frac{dx}{x^2+\frac{1}{x^2}+(a^2-2)} \\ \begin{array}{l}=\underset{0}{\overset{\infty }{\int }}\frac{x^{2}dx}{x^4+(a^2-2)x^2+1}(a^2-2=k\ge 2)\\ =\underset{0}{\overset{\infty }{\int }}\frac{x^{2}dx}{x^4+k{x}^2+1}=\frac{1}{2}\underset{0}{\overset{\infty }{\int }}\frac{(x^2+1)+(x^2-1)}{x^4+k{x}^2+1}dx\\ =\frac{1}{2}\underset{I_1}{\underbrace{\underset{0}{\overset{\infty }{\int }}\frac{1+\left(\frac{1}{x^2}\right)}{x^2+\left(\frac{1}{x^2}\right)+k}}}dx+\frac{1}{2}\underset{I_2}{\underbrace{\underset{0}{\overset{\infty }{\int }}\frac{1-\left(\frac{1}{x^2}\right)}{x^2+\left(\frac{1}{x^2}\right)+k}}}dx\end{array} \\ Nowproceed,I_1=\frac{\pi }{2a}and{I}_1=0 \\ \begin{array}{l}I=\frac{\pi }{2a}=\frac{\pi }{5050}\\ \Rightarrow a=2525=PQRS\\ P+Q+R-S=4\end{array} \end{gathered}$$