Question Number 187898 by cortano12 last updated on 23/Feb/23 $$\:\:\int\:\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{x}}{\mathrm{cos}\:\mathrm{x}+\mathrm{sin}\:\mathrm{x}−\mathrm{1}}\:\mathrm{dx}=? \\ $$ Answered by horsebrand11 last updated on 24/Feb/23 $$\:=\frac{\mathrm{1}}{\mathrm{2}}\int\:\frac{\mathrm{2}−\mathrm{2cos}\:{x}}{\mathrm{cos}\:{x}+\mathrm{sin}\:−\mathrm{1}}\:{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\:\frac{\left(\mathrm{1}−\mathrm{cos}\:{x}+\mathrm{sin}\:{x}\right)−\left(\mathrm{cos}\:{x}+\mathrm{sin}\:{x}−\mathrm{1}\right)}{\mathrm{cos}\:{x}+\mathrm{sin}\:{x}−\mathrm{1}}\:{dx} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{2}}{x}−\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{sin}\:\frac{\mathrm{1}}{\mathrm{2}}{x}+\mathrm{cos}\:\frac{\mathrm{1}}{\mathrm{2}}{x}}{\mathrm{sin}\:\frac{\mathrm{1}}{\mathrm{2}}{x}−\mathrm{cos}\:\frac{\mathrm{1}}{\mathrm{2}}{x}}\:{dx} \\…
Question Number 187890 by Rupesh123 last updated on 23/Feb/23 Answered by aleks041103 last updated on 23/Feb/23 $${I}=\underset{−\mathrm{1}} {\overset{\mathrm{1}} {\int}}\frac{\mathrm{4}\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{\mathrm{1}+\mathrm{2}^{{x}} }{dx} \\ $$$${u}=−{x},\:{du}=−{dx} \\ $$$${I}=\int_{\mathrm{1}}…
Question Number 122349 by liberty last updated on 16/Nov/20 $$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{polynomial}\:\mathrm{P}\left(\mathrm{x}\right)\:\mathrm{of}\:\mathrm{least}\:\mathrm{degree} \\ $$$$\mathrm{that}\:\mathrm{has}\:\mathrm{a}\:\mathrm{maximum}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{6}\:\mathrm{at}\:\mathrm{x}=\mathrm{1} \\ $$$$\mathrm{and}\:\mathrm{minimum}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{2}\:\mathrm{at}\:\mathrm{x}=\mathrm{3}.\: \\ $$ Commented by benjo_mathlover last updated on 17/Nov/20 $${my}\:{answer}\::\: \\…
Question Number 122330 by mnjuly1970 last updated on 15/Nov/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{advanced}\:\:{calculus}… \\ $$$$\:\:{prove}\:{that}\:: \\ $$$$\:\:\:\mathscr{R}{e}\left(\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{3}} \left({x}\right){ln}\left({ln}\left({cos}\left({x}\right)\right)\right){dx}\right) \\ $$$$\:\:\:\:\:\:\:\overset{?} {=}\frac{{ln}\left(\mathrm{3}\right)−\mathrm{2}\gamma}{\mathrm{3}}\:\checkmark \\ $$ Answered by mathmax…
Question Number 122323 by mathmax by abdo last updated on 15/Nov/20 $$\mathrm{calculate}\:\:\mathrm{A}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2}\right)….\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{n}\right)} \\ $$$$\mathrm{wth}\:\mathrm{n}\:\mathrm{integr}\:\mathrm{natural}\:\mathrm{and}\:\mathrm{n}\geqslant\mathrm{1} \\ $$ Commented by Dwaipayan…
Question Number 187855 by Michaelfaraday last updated on 23/Feb/23 $${solve} \\ $$$$\int\frac{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}{\mathrm{2}\left({x}^{\mathrm{2}} +\mathrm{1}\right)}{dx} \\ $$ Answered by cortano12 last updated on 23/Feb/23 $$\:\mathrm{I}=\frac{\mathrm{1}}{\mathrm{2}}\int\:\frac{\left(\mathrm{x}^{\mathrm{2}}…
Question Number 122298 by mnjuly1970 last updated on 15/Nov/20 $$\:\:\:\:\:…{nice}\:\:{calculus}… \\ $$$$\:\:{prove}\:\:{that}\::\:\: \\ $$$$\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}\:} {\overset{\infty} {\sum}}\left\{\frac{\zeta\left(\mathrm{2}{n}+\mathrm{1}\right)−\mathrm{1}}{{n}+\mathrm{1}}\right\}=−\gamma+{ln}\left(\mathrm{2}\right)\checkmark \\ $$$$\:\:\:..{m}.{n}.\mathrm{1970}.. \\ $$ Answered by mindispower last updated…
Question Number 122290 by mnjuly1970 last updated on 15/Nov/20 $$\:\:\:\:\:…\:\:{nice}\:\:{calculus}… \\ $$$$\:\:\:{calculate}\::: \\ $$$$\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{\mathrm{2}} \left(\psi\left(\mathrm{1}+{x}\right)−\psi\left(\mathrm{2}−{x}\right)\right){dx}=??? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:.{m}.{n}.\mathrm{1970}. \\ $$ Commented by Dwaipayan Shikari…
Question Number 56747 by Tawa1 last updated on 22/Mar/19 $$\int\:\mathrm{x}^{\mathrm{2}\:} \mathrm{e}^{\mathrm{x}^{\mathrm{2}} } \:\:\mathrm{dx} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 22/Mar/19 $$\int{x}^{\mathrm{2}} \left(\mathrm{1}+{x}^{\mathrm{2}} +\frac{{x}^{\mathrm{4}}…
Question Number 122274 by benjo_mathlover last updated on 15/Nov/20 $$\:\:\:\int\:\frac{\sqrt{\mathrm{1}−\sqrt{{x}}}}{\:\sqrt{\mathrm{1}+\sqrt{{x}}}}\:{dx}\:?\: \\ $$ Answered by liberty last updated on 15/Nov/20 $$\mathrm{let}\:\sqrt{\mathrm{x}}\:=\:\mathrm{cos}\:\theta\:\Rightarrow\:\mathrm{x}=\mathrm{cos}\:^{\mathrm{2}} \theta\: \\ $$$$\Rightarrow\:\mathrm{dx}\:=−\mathrm{2sin}\:\theta\:\mathrm{cos}\:\theta\:\mathrm{d}\theta\: \\ $$$$\mathrm{I}\:=\int\:\frac{\sqrt{\mathrm{1}−\mathrm{cos}\:\theta}}{\:\sqrt{\mathrm{1}+\mathrm{cos}\:\theta}}\:\left(−\mathrm{2sin}\:\theta\:\mathrm{cos}\:\theta\:\right)\mathrm{d}\theta…