An $18$-year-old $1^\text{st}$ year BS-MS student at IISER Pune. I’m fascinated by math, especially integral calculus. Hence, my username is an ‘integration’ of integral and geek(and not something related to Ελληνικά).

Some of my favourite answers(in no particular order); these are the answers I gained the most insight from while working through the questions:

• Solve $\tan x =\sec 42^\circ +\sqrt{3}$
• Is there a direct derivation to show that $\int_0^\infty \frac{1-x(2-\sqrt x)}{1-x^3}dx$ vanishes?
• What is the physical meaning of sine, cosine and tangent of an obtuse angle?
• Really advanced techniques of integration (definite or indefinite)
• Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$
• Integral $\int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$
• Evaluate $\int\limits^{\infty}_{0} \frac{1}{x} \, \ln\left(\frac{x^{3} - x^{2} - x + 1}{x^{3} + x^{2} + x + 1}\right) \, \mathrm{d}x$
• Finding $\int^\infty_0\frac{\ln^2(x)}{(1-x^2)^2}\mathrm dx$

Some of my answers which I believe didn't get much attention because they were late to the party:

• Find the integral $\int \frac{(\ln(x))^2}{x^3} \, dx$
• Find a primitive of $x^2\sqrt{a^2 - x^2}$
• Help evaluating $\int \frac{dx}{(x^2 + a^2)^2}$
• Any way to solve $\sqrt{x} + \sqrt{x+1} + \sqrt{x+2} = \sqrt{x+7}$?
• Solution of the ODE $\left(\frac{\mathrm dy}{\mathrm dx}\right)^2-x\frac{\mathrm dy}{\mathrm dx}+y=0$
• Evaluation of $\int_{0}^{1}\frac{x\ln (x)}{\sqrt{1-x^2}}dx$

My most favourite identities:

$$\sinh^{-1}x=-i\sin^{-1}ix$$ $$\cosh^{-1}x=i\operatorname{sgn}(1-x^2)\cos^{-1}x~\forall~x\in\Bbb R$$ $$\tanh^{-1}x=-i\tan^{-1}ix$$

A few numerical identities that strike me as beautiful(the last one forms the basis for one of my questions):

$$e^{\pi i}=-1$$ $$\ln\left(\cot\frac{\pi}8\right)=\sinh^{-1}1$$ $$\ln\left(\cot\frac{\pi}{12}\right)=\cosh^{-1}2$$ $$\ln\phi=\operatorname{csch}^{-1}2$$ $$1-\frac13+\frac15-\frac17\cdots=\frac23\cdot\frac43\cdot\frac45\cdot\frac65\cdot\frac67\cdots$$ $$1+\frac{1^2}{3(3!)}+\frac{1^23^2}{5(5!)}+\frac{1^23^25^2}{7(7!)}\cdots=\frac\pi2\ln2$$