I am a first-year CS undergrad at NIT Jalandhar. I have completed a course on Complex Analysis. I have developed interest for Mathematical Modelling of Physics and Geography. My collaborative distance with every mathematician or researcher in general is $int_{1}^{infty} {1}/{x^p}mathrm dx$ such that $p le 1$, but I`m trying hard to push $p$ out of $(-infty, 1]$. $$begin{aligned}&int arctanleft(frac{2cos^2theta}{2-sin(2theta)}right)sec^2theta mathrm dtheta =int (arctan u-arctan(u-1) )mathrm du , u=tantheta end{aligned}$$ Few results compiled: $1$. (Gauss` circle problem) Number of ordered pairs $(x,y), x,yinmathbb{Z}:x^2+y^2le r^2$ is given by $N(r)$. $[.]$ denotes $lfloor xrfloor $. $$boxed{N(r)=1+4sum_{i=0}^{infty}left(left[frac{r^2}{4i+1}right]-left[frac{r^2}{4i+3}right]right)}$$ $2$. Number of points with integer coordinates strictly inside the triangle bounded by $x+y=n$ and the coordinate axes is given by $N(r)$. $$boxed{N(r)=sum_{k=1}^{n-2}(n-k)=sum_{k=1}^{n-2}k=frac{(n-1)(n-2)}{2}}$$ $3.$ (Glasser`s master theorem)(Cauchy–Schlömilch transformation) () If $f(x)$ is a continuous function on $mathbb{R}$ and the line integral of $f(x)$ on $mathbb{R}$ exists and $a gt 0$. $$boxed{int_{-infty}^{+infty}f(x)mathrm dx=int_{-infty}^{+infty}fleft(x-frac{a}{x}right)mathrm dx}$$ $4.$ (Frullani integral) $f(x)$ is a function over $xge 0$ and $lim f(x)$ as $xto infty$ exists, then the Frullani definite integral can be evaluated as follows. $$boxed{int_{0}^{infty}frac{f(ax)-f(bx)}{x}mathrm dx=left(f(0)-lim_{xtoinfty}f(x)right)ln frac{b}{a}}$$ $5$. (Ramanujan`s nested radical) Ramanujan gave the general result for the general nested radical which holds for all $a$, $n$ and $x$ in $mathbb{R}$. $$boxed{sqrt{ax+(n+a)^2+xsqrt{a(x+n)+(n+a)^2+(x+n)sqrt{.}}}=x+n+a}$$ $6$. (Herschfeld`s convergence theorem) For real, nonnegative terms $x_n$ and real $p$ with $0<p<1$, the following expressions converges iff $exists Minmathbb{R} Mge(x_{n})^{p^{n}}$. $$boxed{lim_{kto infty}x_{0}+(x_{1}+(x_{2}+(ldots +(x_{k})^{p})^{p})^{p})^{p}}$$ ©