Question Number 136813 by mnjuly1970 last updated on 26/Mar/21 $$\:\:\:\:\: \\ $$$$\:\:\:……{advanced}\:\:\:\:{calculus}\left({I}\right)…. \\ $$$$\:\:\:\:{if}\::\: \\ $$$$\:\:\Omega=\int_{\frac{\pi}{\mathrm{4}}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{x}.{cos}\left(\mathrm{2}{x}\right).{cos}\left({x}\right)}{{sin}^{\mathrm{7}} \left({x}\right)}\:{dx}=\frac{{a}\pi}{\mathrm{90}} \\ $$$$\:\:\:{then}\:\:::\:\:{a}=??? \\ $$$$\:\: \\ $$$$\:\:\:\:…
Question Number 136806 by mnjuly1970 last updated on 26/Mar/21 Answered by Dwaipayan Shikari last updated on 26/Mar/21 $$\mathrm{2}\int_{\mathrm{0}} ^{\infty} \frac{{sin}\left(\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\right)}{\:\sqrt[{\mathrm{3}}]{{x}}}{dx}\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{{x}^{\mathrm{3}} }={u}\Rightarrow−\frac{\mathrm{3}}{{x}^{\mathrm{4}} }=\frac{{du}}{{dx}} \\ $$$$=\frac{\mathrm{2}}{\mathrm{3}}\int_{\mathrm{0}}…
Question Number 136803 by mnjuly1970 last updated on 26/Mar/21 $$\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:…..\mathscr{A}{dvanced}\:\:\:\:\blacktriangleleft………….\blacktriangleright\:\:\:\mathscr{C}{alculus}….. \\ $$$$\:\:\:\:\:\boldsymbol{\phi}=\int_{−\infty} ^{\:\infty} \frac{{sin}\left(\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\right)}{{x}}{dx}=……??? \\ $$$$\:\:\:\:\:{solution}:: \\ $$$$\:\:\:\:\:\:\boldsymbol{\phi}\overset{\frac{\mathrm{1}}{{x}}\:={t}} {=}\:\mathrm{2}\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left({t}^{\mathrm{3}} \right)}{\frac{\mathrm{1}}{{t}}}.\frac{{dt}}{{t}^{\mathrm{2}}…
Question Number 71239 by 20190927 last updated on 13/Oct/19 $$\int\frac{\mathrm{1}}{\mathrm{2cosx}−\mathrm{5sinx}−\mathrm{3}}\mathrm{dx} \\ $$ Commented by mathmax by abdo last updated on 13/Oct/19 $${let}\:{I}\:=\int\:\:\:\:\frac{{dx}}{\mathrm{2}{cosx}−\mathrm{5}{sinx}\:−\mathrm{3}}\:\:{changement}\:{tan}\left(\frac{{x}}{\mathrm{2}}\right)={t}\:{give} \\ $$$${I}\:=\int\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}\frac{\mathrm{1}−{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}}…
Question Number 136762 by snipers237 last updated on 25/Mar/21 $$\:{let}\:\:{C}\left({a},{r}\right)=\left\{{z}\in\mathbb{C},\:\:\mid{z}−{a}\mid={r}\:\right\} \\ $$$${Let}\:{u},{v},{w}\:\in{C}\left({a},{r}\right)\:{such}\:{as}\:\:\:{u}+{v}=\mathrm{2}{w} \\ $$$${Prove}\:{that}\:\:\frac{{u}−{a}}{{v}−{a}}\:=\mathrm{1}\:,\:{u}={v}={w} \\ $$$$ \\ $$$${It}\:{shows}\:{that}\:{the}\:{middle}\:{of}\:{a}\:{segment}\: \\ $$$${joining}\:{two}\:{points}\:{in}\:{a}\:{circle}\:\:{is}\:{not}\:{in}\:{that}\:{circle} \\ $$ Terms of Service…
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Question Number 136750 by mnjuly1970 last updated on 25/Mar/21 $$\:\:\:\:\:\:\:\:\:\:\:…..{advanced}\:\:\:\:{calculus}….. \\ $$$$\:\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:…\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{1}−{e}^{−{x}^{\mathrm{2}} } }{{x}^{\mathrm{2}} }{dx}=\sqrt{\pi}\:\:\:\: \\ $$ Answered by Ñï= last…
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Question Number 136723 by mnjuly1970 last updated on 25/Mar/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….{calculus}\:\:\:\:\:\left({I}\right)….. \\ $$$$\:\:\:\:\:{prove}\:\:\:{that}\::: \\ $$$$\:\:\:\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−\mathrm{4}{x}}}\:\overset{???} {=}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\begin{pmatrix}{\:\mathrm{2}{n}}\\{\:\:\:{n}}\end{pmatrix}\:{x}^{{n}} \\ $$$$ \\ $$ Answered by Dwaipayan Shikari…
Question Number 71175 by 20190927 last updated on 12/Oct/19 $$\int\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} +\mathrm{2016x}}\mathrm{dx} \\ $$ Commented by Prithwish sen last updated on 12/Oct/19 $$\int\frac{\mathrm{dx}}{\left(\mathrm{x}+\mathrm{1008}\right)^{\mathrm{2}} −\left(\mathrm{1008}\right)^{\mathrm{2}} }\:\:\:\:\:\:\boldsymbol{\mathrm{put}}\:\boldsymbol{\mathrm{x}}+\mathrm{1008}\:=\:\boldsymbol{\mathrm{t}} \\…