Question Number 20245 by virus last updated on 24/Aug/17 $$\mathrm{y}=\mathrm{2}^{\frac{\mathrm{1}}{{log}_{{x}} \mathrm{8}}} \\ $$$${then}\:{x}=? \\ $$ Answered by Tinkutara last updated on 24/Aug/17 $${y}\:=\:\mathrm{2}^{\mathrm{log}_{\mathrm{8}} \:{x}} \:=\:\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{3}}\:\mathrm{log}_{\mathrm{2}}…
Question Number 85781 by Joel578 last updated on 24/Mar/20 $$\int_{\mathrm{0}} ^{\mathrm{1}} \:\left(\mathrm{ln}\:\frac{\mathrm{1}}{{x}}\right)^{−\mathrm{3}/\mathrm{2}} \:{dx} \\ $$ Commented by Joel578 last updated on 24/Mar/20 $$\mathrm{If}\:\mathrm{we}\:\mathrm{use}\:{u}\:=\:\mathrm{ln}\:\frac{\mathrm{1}}{{x}},\:\mathrm{the}\:\mathrm{integral}\:\mathrm{will}\:\mathrm{become} \\ $$$$\int_{\mathrm{0}}…
Question Number 20244 by tammi last updated on 24/Aug/17 $$\int\frac{{dx}}{\left(\mathrm{2}−{x}\right)\sqrt{{x}}} \\ $$ Answered by ajfour last updated on 24/Aug/17 $${let}\:\sqrt{{x}}={t}\:\:\:\:\Rightarrow\:\:\:{dx}=\mathrm{2}{tdt} \\ $$$$\int\frac{{dx}}{\left(\mathrm{2}−{x}\right)\sqrt{{x}}}=\int\frac{\mathrm{2}{tdt}}{\left(\mathrm{2}−{t}^{\mathrm{2}} \right){t}} \\ $$$$=−\mathrm{2}\int\frac{{dt}}{{t}^{\mathrm{2}}…
Question Number 20243 by tammi last updated on 24/Aug/17 $$\int\frac{{dx}}{{x}+\sqrt{{x}}} \\ $$ Answered by $@ty@m last updated on 25/Aug/17 $${Here}\:{I}=\int\frac{\frac{\mathrm{1}}{\:\sqrt{{x}}}}{\:\sqrt{{x}}+\mathrm{1}}{dx}\:\left({dividing}\:{N}^{{r}} \:\&\:{D}^{{r}} \:{by}\:\sqrt{{x}}\right) \\ $$$${Let}\:\sqrt{{x}}+\mathrm{1}={t} \\…
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Question Number 20242 by tammi last updated on 24/Aug/17 $$\int\frac{\mathrm{5}{x}^{\mathrm{2}} +\mathrm{11}{x}+\mathrm{26}}{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}}{dx} \\ $$$${integration}\:{by}\:{partial}\:{fraction} \\ $$ Answered by ajfour last updated on 24/Aug/17 $$\frac{\mathrm{5}{x}^{\mathrm{2}} +\mathrm{11}{x}+\mathrm{26}}{{x}^{\mathrm{2}}…
Question Number 151315 by mathdanisur last updated on 20/Aug/21 $$\mathrm{if}\:\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =\mathrm{16}^{\mathrm{2}} \:\:;\:\:\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} =\mathrm{24}^{\mathrm{2}} \\ $$$$\mathrm{z}^{\mathrm{2}} +\mathrm{t}^{\mathrm{2}} =\mathrm{42}^{\mathrm{2}} \:\:\mathrm{and}\:\:\mathrm{t}^{\mathrm{2}} +\mathrm{x}^{\mathrm{2}} =\mathrm{38}^{\mathrm{2}} \\ $$$$\mathrm{find}\:\:\mathrm{max}\left[\left(\mathrm{x}+\mathrm{z}\right)\left(\mathrm{y}+\mathrm{t}\right)\right]=? \\…
Question Number 85776 by M±th+et£s last updated on 24/Mar/20 Answered by MJS last updated on 24/Mar/20 $$\int\frac{{a}\:\mathrm{sin}^{\mathrm{2}} \:{x}\:+\mathrm{2}{b}\:\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}\:+{c}\:\mathrm{cos}^{\mathrm{2}} \:{x}}{\mathrm{sin}\:{x}\:+\mathrm{cos}\:{x}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\mathrm{tan}\:\frac{{x}}{\mathrm{2}}\:\rightarrow\:{dx}=\frac{\mathrm{2}}{{t}^{\mathrm{2}} +\mathrm{1}}{dt}\right] \\ $$$$=−\mathrm{2}\int\frac{{ct}^{\mathrm{4}} −\mathrm{4}{bt}^{\mathrm{3}}…
Question Number 20241 by tammi last updated on 24/Aug/17 $${partial}\:{fraction} \\ $$$$\int\frac{\mathrm{2}{x}^{\mathrm{2}} +\mathrm{5}{x}−\mathrm{9}}{\:\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}}{dx} \\ $$ Answered by $@ty@m last updated on 25/Aug/17 $$=\int\frac{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{2}+\mathrm{7}{x}−\mathrm{7}−\mathrm{4}}{\:\sqrt{{x}^{\mathrm{2}}…
Question Number 85774 by M±th+et£s last updated on 24/Mar/20 $$\left({x}_{\mathrm{2}{n}} \right)=\mathrm{2}^{\mathrm{2}{n}} \left(\frac{{x}}{\mathrm{2}}\right)_{{n}} \left(\frac{{x}+\mathrm{1}}{\mathrm{2}}\right)_{{n}} \\ $$$$\left({x}\right)_{{m}\:{n}} ={m}^{{m}\:{n}} \underset{{k}=\mathrm{0}} {\overset{{m}=\mathrm{1}} {\prod}}\left(\frac{{x}+{k}}{{m}}\right)_{{n}} \:\:\:,\:{m}\in{z} \\ $$$${now}\:{if}\:{m}\:{is}\:{relative}\:{number}\:{such}\:{as}\frac{\mathrm{3}}{\mathrm{2}}\:,\:{m}\in{Q} \\ $$$$\left({x}\right)_{\frac{\mathrm{3}}{\mathrm{2}}{n}} =??…