Question Number 64037 by MJS last updated on 18/Nov/19 $$\mathrm{reduction}\:\mathrm{formulas}\:\mathrm{for}\:{n}\in\mathbb{N},\:\mathrm{some}\:{n}>\mathrm{0},\:\mathrm{some}\:{n}>\mathrm{1} \\ $$$$ \\ $$$$\int\mathrm{sin}^{{n}} \:{x}\:{dx}=−\frac{\mathrm{1}}{{n}}\mathrm{cos}\:{x}\:\mathrm{sin}^{{n}−\mathrm{1}} \:{x}\:+\frac{{n}−\mathrm{1}}{{n}}\int\mathrm{sin}^{{n}−\mathrm{2}} \:{x}\:{dx} \\ $$$$\int\mathrm{cos}^{{n}} \:{x}\:{dx}=\frac{\mathrm{1}}{{n}}\mathrm{sin}\:{x}\:\mathrm{cos}^{{n}−\mathrm{1}} \:{x}\:+\frac{{n}−\mathrm{1}}{{n}}\int\mathrm{cos}^{{n}−\mathrm{2}} \:{x}\:{dx} \\ $$$$\int\mathrm{tan}^{{n}} \:{x}\:{dx}=\frac{\mathrm{1}}{{n}−\mathrm{1}}\mathrm{tan}^{{n}−\mathrm{1}}…
Question Number 129562 by mathmax by abdo last updated on 16/Jan/21 $$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{6}} +\mathrm{1}\right)^{\mathrm{2}} } \\ $$ Answered by Lordose last updated on 16/Jan/21…
Question Number 129564 by bramlexs22 last updated on 16/Jan/21 $$\:\mathcal{V}\:=\:\int\:\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{sin}\:\left(\mathrm{x}+\theta\right)}\:\mathrm{dx}\: \\ $$ Answered by Lordose last updated on 16/Jan/21 $$\Omega\:=\:\int\frac{\mathrm{sin}\left(\mathrm{x}\right)}{\mathrm{sin}\left(\mathrm{x}+\theta\right)}\mathrm{dx}\:\overset{\mathrm{u}=\mathrm{x}+\theta} {=}\int\frac{\mathrm{sin}\left(\mathrm{u}−\theta\right)}{\mathrm{sin}\left(\mathrm{u}\right)}\mathrm{du} \\ $$$$\Omega\:=\:\int\frac{\mathrm{sin}\left(\mathrm{u}\right)\mathrm{cos}\theta−\mathrm{cos}\left(\mathrm{u}\right)\mathrm{sin}\theta}{\mathrm{sin}\left(\mathrm{u}\right)}\mathrm{du}\:=\:\mathrm{ucos}\theta\:−\:\mathrm{sin}\theta\mathrm{ln}\left(\mathrm{sinu}\right)\:+\:\mathrm{C} \\ $$$$\Omega\:=\:\left(\mathrm{x}+\theta\right)\mathrm{cos}\theta\:−\:\mathrm{sin}\theta\mathrm{ln}\left(\mathrm{sin}\left(\mathrm{x}+\theta\right)\right)+\:\mathrm{C}…
Question Number 129558 by mnjuly1970 last updated on 16/Jan/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{modern}\:\ast\ast\ast\ast\ast\ast\ast\ast\ast\ast\:{algebra}\:…\: \\ $$$$\:\:\:\:\:\:\:\::::\:\:{if}\:\:''\:{G}\:''\:{be}\:{a}\:{finite}\:{group}\:{and} \\ $$$$\:\:{O}\:\left({G}\right)={pq}\:\:,\:\:{where}\:''\:{p}\:,\:{q}\:''\:{are}\:{two} \\ $$$$\:\:{prime}\:\:{numbers}\:\left({p}\:>\:{q}\:\right)\:{then}\:{prove}\:{that}: \\ $$$$\:\:{G}\:\:{has}\:\:{at}\:{most}\:{one}\:{subgroup}\:{of}\:{order}\:''\:{p}\:''\:. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{written}\:{and}\:{compiled}\:{by} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…\clubsuit{m}.{n}.{july}.\mathrm{1970}\clubsuit…. \\ $$ Answered…
Question Number 129519 by BHOOPENDRA last updated on 16/Jan/21 $$\int\:{cos}\:\left({y}^{\mathrm{3}} \right){dy} \\ $$ Answered by Dwaipayan Shikari last updated on 16/Jan/21 $$\int{cos}\left({y}^{\mathrm{3}} \right){dy} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int{e}^{{iy}^{\mathrm{3}}…
Question Number 63976 by Scientist0000001 last updated on 11/Jul/19 $$\int{secxdx}\:\:\:\:? \\ $$ Commented by Prithwish sen last updated on 12/Jul/19 $$\int\frac{\mathrm{sec}^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}}{\mathrm{1}−\mathrm{tan}^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}}\mathrm{dx}\:\:\:\:\mathrm{putting}\:\mathrm{tan}\frac{\mathrm{x}}{\mathrm{2}}\:=\:\mathrm{t} \\ $$$$\mathrm{sec}^{\mathrm{2}}…
Question Number 129503 by BHOOPENDRA last updated on 16/Jan/21 Commented by talminator2856791 last updated on 16/Jan/21 $$\:\mathrm{is}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{half}\:\mathrm{the}\:\mathrm{sqhere}? \\ $$ Commented by BHOOPENDRA last updated on…
Question Number 63927 by aliesam last updated on 11/Jul/19 $$\int_{\mathrm{0}} ^{\pi} \frac{{dx}}{\left(\mathrm{3}+\mathrm{2}{cos}\:{x}\right)^{\mathrm{2}} } \\ $$ Commented by aliesam last updated on 12/Jul/19 $${god}\:{bless}\:{you}\:{sir}\:..{well}\:{done}.. \\ $$…
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Question Number 63892 by mathmax by abdo last updated on 10/Jul/19 $${calculate}\:{A}=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{2017}} }{\mathrm{1}+{x}^{\mathrm{2019}} }\:{dx}\:\:{and}\:{B}\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{\mathrm{2019}} }{\mathrm{1}+{x}^{\mathrm{2021}} }\:{dx} \\ $$$${calculate}\:{the}\:{fraction}\:\frac{{A}}{{B}} \\ $$ Commented…