Area 51 Proposal: Math Challenges! If you're interested, please follow and upvote $5$ example questions that haven't reached a score of $10$ yet!

Chatroom for proposal and meta post.

My WordPress blog is here!

About: Where I store any mathematical problems I've made up :)

My favourite questions:

• Finding the turning points of $f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x$

• Some interesting observations on a sum of reciprocals

• Area of a mushroom-shaped curve

• How to show that $\int_0^\infty\frac1{x^x}\,dx<2$

• Mathematical coincidences concerning the numbers $\pi$, $e$ and $163$

• Convergence concerning the $\alpha$th derivative of $f(x)=x^{\alpha}-\alpha^x$

My favourite answers:

• Evaluating $\int_0^1\frac{3x^4+ 4x^3 + 3x^2}{(4x^3 + 3x^2 + 2x+ 1)^2}\, dx$

• How to prove :$\sqrt{1!\sqrt{2!\sqrt{3!\sqrt{\cdots\sqrt{n!}}}}} <3$

• Evaluate $\int (1-x^{2008})^{\frac{1}{2007}} (1-x^{2007})^{\frac{1}{2008}} dx$

• General formula for arctan

Just discovered a neat closed form of the following integral using $1$s and $2$s: $$\begin{align}\int_{\pi/8}^{\pi/4}\frac{\sin x+\cos x}{\tan x}\,dx&=\sqrt2-\frac{\sqrt{\sqrt2-1}+\sqrt{\sqrt2+1}}{\sqrt{2\sqrt2}}\\&\,\,\,\,\,\,+\ln\left(1+\sqrt2\sqrt{2-\sqrt2}\right)\end{align}$$