Question Number 93860 by Ar Brandon last updated on 15/May/20 $$\mathrm{Consider}\:\mathrm{the}\:\mathrm{progression}\:\left(\mathrm{I}_{\mathrm{n}} \right)_{\mathrm{n}\in\mathbb{N}} \:\mathrm{where}\:\mathrm{I}_{\mathrm{n}} \:=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{sin}\left(\pi\mathrm{t}\right)}{\mathrm{t}+\mathrm{n}}\mathrm{dt} \\ $$$$\mathrm{1}\backslash\:\mathrm{Show}\:\mathrm{that}:\:\forall\mathrm{n}\geqslant\mathrm{0},\:\mathrm{0}\leqslant\mathrm{I}_{\mathrm{n}+\mathrm{1}} \leqslant\mathrm{I}_{\mathrm{n}} \:\mathrm{and}\:\mathrm{deduce}\:\mathrm{that}\:\mathrm{the}\: \\ $$$$\mathrm{series}\:\mathrm{is}\:\mathrm{convergent}. \\ $$$$\mathrm{2}\backslash\:\mathrm{Show}\:\mathrm{that}\:\mathrm{0}\leqslant\mathrm{I}_{\mathrm{n}} \leqslant\mathrm{ln}\left(\frac{\mathrm{n}+\mathrm{1}}{\mathrm{n}}\right)\:\mathrm{and}\:\mathrm{deduce}\:\mathrm{the}\:\mathrm{limit}…
Question Number 93820 by i jagooll last updated on 15/May/20 $$\int\:\frac{{x}^{\mathrm{2}} \:{dx}}{\:\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}}\: \\ $$ Commented by mathmax by abdo last updated on 17/May/20 $${I}\:=\int\:\:\frac{{x}^{\mathrm{2}}…
Question Number 93804 by i jagooll last updated on 15/May/20 $$\int\:{x}\:\sqrt[{\mathrm{3}\:\:}]{\frac{\mathrm{3}{x}−\mathrm{1}}{{x}+\mathrm{2}}}\:{dx}\:?\: \\ $$ Commented by i jagooll last updated on 15/May/20 $$\mathrm{set}\:\mathrm{t}^{\mathrm{3}} \:=\:\frac{\mathrm{3x}−\mathrm{1}}{\mathrm{x}+\mathrm{2}}\:\Rightarrow\:\mathrm{xt}^{\mathrm{3}} +\mathrm{2t}^{\mathrm{3}} =\mathrm{3x}−\mathrm{1}\:…
Question Number 28268 by abdo imad last updated on 22/Jan/18 $$\:{find}\:{in}\:{terms}\:{of}\:{n}\:{the}\:{value}\:{of} \\ $$$${A}_{{n}} =\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{{n}}{\mathrm{2}}} \:{sin}\left({narctanx}\right){dx}\:.\:\:\left(\:{n}\in\:{N}\right). \\ $$ Commented by abdo imad last…
Question Number 28263 by abdo imad last updated on 22/Jan/18 $${decompose}\:{F}\left({x}\right)=\:\:\frac{\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} \left({x}^{\mathrm{3}} −\mathrm{1}\right)}\:\:{then}\:{calculate} \\ $$$$\int_{\mathrm{2}} ^{+\propto} \:{F}\left({x}\right){dx}. \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 28262 by abdo imad last updated on 22/Jan/18 $${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{5}} }\:. \\ $$ Commented by abdo imad last updated on 24/Jan/18 $${let}\:{put}\:{x}^{\mathrm{5}}…
Question Number 28247 by abdo imad last updated on 22/Jan/18 $${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{lnx}}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:. \\ $$ Commented by abdo imad last updated on 24/Jan/18 $${let}\:{put}\:\:{I}=\:\int_{\mathrm{1}}…
Question Number 28242 by abdo imad last updated on 22/Jan/18 $${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} {e}^{−{x}} {lnxdx}\:\:. \\ $$ Commented by abdo imad last updated on 23/Jan/18 $${let}\:{put}\:{I}_{{n}}…
Question Number 159315 by cortano last updated on 15/Nov/21 $$\:\int\:\frac{{dx}}{\:\sqrt[{\mathrm{4}}]{\left({x}−\mathrm{1}\right)^{\mathrm{3}} \left({x}+\mathrm{2}\right)^{\mathrm{5}} }\:}\:? \\ $$ Commented by tounghoungko last updated on 15/Nov/21 $${Y}=\int\:\frac{{dx}}{\left({x}+\mathrm{2}\right)^{\mathrm{2}} \left(\frac{{x}−\mathrm{1}}{{x}+\mathrm{2}}\right)^{\mathrm{3}/\mathrm{4}} }\: \\…
Question Number 93762 by ckkim89 last updated on 14/May/20 Answered by maths mind last updated on 14/May/20 $$=\frac{\mathrm{4}{xe}^{\mathrm{2}{x}} }{\mathrm{4}\left(\mathrm{1}+\mathrm{2}{x}\right)^{\mathrm{2}} }=\frac{\mathrm{1}}{\mathrm{4}}.\frac{{de}^{\mathrm{2}{x}} .\left(\mathrm{1}+\mathrm{2}{x}\right)−{d}\left(\mathrm{1}+\mathrm{2}{x}\right).{e}^{\mathrm{2}{x}} }{\left(\mathrm{1}+\mathrm{2}{x}\right)^{\mathrm{2}} } \\ $$$$\Rightarrow\int\frac{{xe}^{\mathrm{2}{x}}…