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TheSimpliFire

AREA 51 PROPOSAL: MATH CHALLENGES! You're more than welcome to follow if interested.

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


Challenge: (01/09 - 15/09)

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Find the value of $$\small\int_1^{\sqrt e}\frac{(x^2-1)(\ln x+1)^2+x(\ln x-1)^2-(3x-1)(x+2)}{(1-x\ln x)^2}\,dx.$$


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

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