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x-arctanxdx-1-x-2-1-k-2-x-2-1-k-2-F-pi-4-k-pi-2-2-1-k-2-

Question Number 223034 by MrGaster last updated on 13/Jul/25 $$ \\ $$$$\int\frac{{x}\:\mathrm{arctan}{xdx}}{\:\sqrt{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{k}'^{\mathrm{2}} {x}^{\mathrm{2}} \right)}}=\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\left[{F}\left(\frac{\pi}{\mathrm{4}},{k}\right)−\frac{\pi}{\mathrm{2}\sqrt{\mathrm{2}\left(\mathrm{1}+{k}'\mathrm{2}\right)}}\right] \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com

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Question Number 223049 by wewji12 last updated on 13/Jul/25 $$\mathrm{we}\:\mathrm{already}\:\mathrm{know}\:\:\underset{{h}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{h}}=\infty\:\mathrm{and}\:\underset{{p}\in\mathbb{P}} {\sum}\:\frac{\mathrm{1}}{{p}}=\infty \\ $$$$\mathrm{from}\:\mathrm{the}\:\mathrm{Harmonic}\:\mathrm{Series} \\ $$$$\underset{{h}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{h}}=\underset{{p}\in\mathbb{P}} {\prod}\:\left(\mathrm{1}+\frac{\mathrm{1}}{{p}}+\left(\frac{\mathrm{1}}{{p}}\right)^{\mathrm{2}} +\left(\frac{\mathrm{1}}{{p}}\right)^{\mathrm{3}} +…\right)=\:\underset{{p}\in\mathbb{P}} {\prod}\:\frac{\mathrm{1}}{\mathrm{1}−{p}^{−\mathrm{1}} } \\…

Question-223040

Question Number 223040 by MrGaster last updated on 13/Jul/25 Commented by Nicholas666 last updated on 14/Jul/25 $$\:\:\mathrm{thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{sir},\: \\ $$$$\:\mathrm{very}\:\mathrm{nice}\:\mathrm{you}'\mathrm{re}\:\mathrm{approach}\:\:\:\mathrm{solution}\:\mathrm{to}\:\mathrm{this}\:\mathrm{problem}\: \\ $$$$ \\ $$ Terms of…