Question Number 135174 by bemath last updated on 11/Mar/21 $$\mathcal{Z}\:=\:\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{arctan}\:\left(\mathrm{sin}\:\mathrm{x}\right)\:\mathrm{dx}\:+\:\int_{\mathrm{0}} ^{\:\pi/\mathrm{4}} \mathrm{arcsin}\:\left(\mathrm{tan}\:\mathrm{x}\right)\:\mathrm{dx} \\ $$ Answered by john_santu last updated on 11/Mar/21 $${let}\:\mathcal{Z}_{\mathrm{1}} =\int_{\mathrm{0}}…
Question Number 69637 by MJS last updated on 26/Sep/19 $$…\mathrm{now}\:\mathrm{try}\:\mathrm{this}\:\mathrm{one}: \\ $$$$\int\frac{{dx}}{{x}^{\mathrm{1}/\mathrm{2}} −{x}^{\mathrm{1}/\mathrm{3}} −{x}^{\mathrm{1}/\mathrm{6}} }= \\ $$ Answered by Kunal12588 last updated on 26/Sep/19 $${t}={x}^{\mathrm{1}/\mathrm{6}}…
Question Number 4100 by 123456 last updated on 28/Dec/15 $$\mathrm{lets}\:{f}\:\mathrm{continuous}\:\mathrm{and}\:\mathrm{diferrenciable} \\ $$$${f}\left({x}+\mathrm{1}\right)={xf}\left({x}\right) \\ $$$$\mathrm{proof}\:\mathrm{that}\:{n}\in\mathbb{N}/\left\{\mathrm{0}\right\} \\ $$$${f}^{\left({n}\right)} \left({x}+\mathrm{1}\right)={nf}^{\left({n}−\mathrm{1}\right)} \left({x}\right)+{xf}^{\left({n}\right)} \left({x}\right) \\ $$$$\mathrm{where} \\ $$$${f}^{\left({n}\right)} \left({x}\right)=\frac{{d}^{{n}} {f}}{{dx}^{{n}}…
Question Number 135169 by bramlexs22 last updated on 11/Mar/21 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\: \\ $$$$\mathrm{x}^{\mathrm{81}} +\mathrm{x}^{\mathrm{49}} +\mathrm{x}^{\mathrm{25}} +\:\mathrm{x}^{\mathrm{9}} +\:\mathrm{x}\:\mathrm{is}\:\mathrm{divided} \\ $$$$\mathrm{by}\:\mathrm{x}^{\mathrm{3}} +\mathrm{x}\: \\ $$ Terms of Service Privacy…
Question Number 135170 by bramlexs22 last updated on 11/Mar/21 $$\mathrm{1}+\frac{\mathrm{2}}{\mathrm{3}}+\frac{\mathrm{3}}{\mathrm{3}^{\mathrm{2}} }+\frac{\mathrm{4}}{\mathrm{3}^{\mathrm{3}} }+\frac{\mathrm{5}}{\mathrm{3}^{\mathrm{4}} }+\frac{\mathrm{6}}{\mathrm{3}^{\mathrm{5}} }+…\:=?\: \\ $$$$ \\ $$ Answered by bemath last updated on 11/Mar/21…
Question Number 135165 by Dwaipayan Shikari last updated on 10/Mar/21 $${Solve}\:{Brachistochrone}\:{Curve}\:{Problem} \\ $$ Commented by Dwaipayan Shikari last updated on 10/Mar/21 $${I}\:{have}\:{a}\:{combined}\:{solution}.\:{But}\:{i}\:{want}\:{to}\:{know}\:{others} \\ $$$$\left.{Solutions}\::\right) \\…
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Question Number 4092 by Filup last updated on 28/Dec/15 $$\mathrm{Prove}: \\ $$$$\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{n}!}{\left({n}−{k}\right)!\:{k}!}=\mathrm{2}^{{n}} −\mathrm{1} \\ $$ Answered by Yozzii last updated on 28/Dec/15 $${According}\:{to}\:{the}\:{Binomial}\:{Theorem},…
Question Number 69623 by aliesam last updated on 25/Sep/19 $$\int\frac{\mathrm{1}}{\:\sqrt{{x}}\:+\:\sqrt[{\mathrm{3}}]{{x}}}\:{dx} \\ $$ Answered by MJS last updated on 25/Sep/19 $$\mathrm{we}\:\mathrm{had}\:\mathrm{this}\:\mathrm{before}… \\ $$$$\int\frac{{dx}}{{x}^{\mathrm{1}/\mathrm{2}} +{x}^{\mathrm{1}/\mathrm{3}} }= \\…
Question Number 69620 by aseer imad last updated on 25/Sep/19 Commented by kaivan.ahmadi last updated on 26/Sep/19 $${b}\:{is}\:{answer}\:{since}\: \\ $$$$\frac{\mathrm{1}}{\mathrm{5}}\left(\mathrm{4},\mathrm{0},−\mathrm{3}\right).\left(\mathrm{3},−\mathrm{1},\mathrm{4}\right)=\frac{\mathrm{1}}{\mathrm{5}}\left(\mathrm{12}+\mathrm{0}−\mathrm{12}\right)=\mathrm{0}\Rightarrow\overset{\rightarrow} {{b}}\:{is} \\ $$$${perpendicular}\:{to}\:\mathrm{3}{i}−{j}+\mathrm{4}{k} \\ $$$${and}…