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### Show an Inequality $\int_{0}^\infty\frac{\sqrt x\ln(1+x+x^2)}{1+x^2}<(1+\sqrt 3)\pi$

Here is to explicitly evaluate the integral, via $\int_0^\infty \frac{\ln(a^2+x^2)}{b^2+x^2}dx=\frac \pi b\ln(a+b)$ \begin{align} & \int_{0}^\infty\frac{\sqrt x\ln(1+x+x^2)}{1+x^2}\overset{x\to x^...
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### Show an Inequality $\int_{0}^\infty\frac{\sqrt x\ln(1+x+x^2)}{1+x^2}<(1+\sqrt 3)\pi$

$$\int_0^\infty\frac{\sqrt x\ln(1+x+x^2)}{1+x^2}dx=\int_{0}^\infty\frac{\sqrt x\ln x}{1+x^2}dx+\int_{0}^\infty\frac{\ln(x+\frac1x+1)}{\sqrt x(x+\frac1x)}dx=I_1+I_2\tag{0}$$ Using the keyhole contour ...
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### Is this unified triangle inequality for sides and angles $\frac{a}{A} + \frac{b}{B} > \frac{c}{C}$ true?

Note that by normalizing, we can set $c=1$. By sine law, we have $a=\dfrac{\sin A}{\sin C}$. Therefore, rearranging we have the statement is equivalent to \dfrac{\sin A}{A}+\dfrac{\sin B}{B}-\dfrac{\...
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