A $17$-year old high school graduate who’s looking forward to joining a college. 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):

• 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 $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$
• Evaluate $\int\limits^{\infty}_{0} \frac{1}{x} \, \ln\left(\frac{x^{3} - x^{2} - x + 1}{x^{3} + x^{2} + x + 1}\right) \, \mathrm{d}x$
• Evaluating $\int_0^1 \frac{\ln(1+x)}{(1+x)(1+x^2)} \mathrm dx$
• 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 identity forms the basis for one of my questions):

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