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Category: Integration

0-dx-x-1-x-2-

Question Number 124217 by bramlexs22 last updated on 01/Dec/20 $$\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{{dx}}{{x}\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:? \\ $$ Answered by Dwaipayan Shikari last updated on 01/Dec/20 $$\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:={t}\Rightarrow\frac{{x}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}}…

nice-calculus-prove-that-m-n-1-1-n-m-n-2-m-2-pi-2-12-pi-4-ln-2-

Question Number 124202 by mnjuly1970 last updated on 01/Dec/20 $$\:\:\:\:\:\:\:\:\:\::::\:\:{nice}\:\:{calculus}\:::: \\ $$$$\:\:\:\:\:{prove}\:\:{that}\:::: \\ $$$$ \\ $$$$\underset{{m},{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left\{\frac{\left(−\mathrm{1}\right)^{{n}+{m}} }{{n}^{\mathrm{2}} +{m}^{\mathrm{2}} }\right\}\:\overset{???} {=}\frac{\pi^{\mathrm{2}} }{\mathrm{12}}\:−\frac{\pi}{\mathrm{4}}{ln}\left(\mathrm{2}\right) \\ $$…

nice-calculus-please-prove-0-x-1-2-x-2-2x-5-dx-pi-where-is-Golden-ratio-

Question Number 124201 by mnjuly1970 last updated on 01/Dec/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\::::\:\:{nice}\:\:{calculus}\:::: \\ $$$$\:\:\:\:\:{please}\:\:{prove}\:::: \\ $$$$\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{{x}^{\frac{\mathrm{1}}{\mathrm{2}}} }{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}}{dx}=\frac{\pi}{\:\sqrt{\varphi}} \\ $$$$\:\:\:\:\:{where}\:\:\varphi\:\:{is}\:{Golden}\:{ratio}… \\ $$ Answered by ajfour…

Question-58639

Question Number 58639 by rahul 19 last updated on 26/Apr/19 Answered by tanmay last updated on 27/Apr/19 $${F}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} {tf}\left({t}\right){dt} \\ $$$$\frac{{dF}}{{dx}}=\int_{\mathrm{0}} ^{{x}} \frac{\partial}{\partial{x}\:}\left({tf}\left({t}\right){dt}\:+{xf}\left({x}\right)\frac{{dx}}{{dx}}−\mathrm{0}×{f}\left(\mathrm{0}\right)×\frac{{d}\mathrm{0}}{{dx}}\right. \\…