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Category: Integration

Question-133470

Question Number 133470 by rs4089 last updated on 22/Feb/21 Answered by mr W last updated on 22/Feb/21 $${x}={a}\:\mathrm{cos}\:\theta \\ $$$${y}={b}\:\mathrm{sin}\:\theta \\ $$$${dV}=−\mathrm{2}\pi\left(\mathrm{2}{a}−{a}\:\mathrm{cos}\:\theta\right)\mathrm{2}{ydx} \\ $$$$=\mathrm{4}\pi{a}\left(\mathrm{2}−\mathrm{cos}\:\theta\right){b}\mathrm{sin}\:\theta{a}\mathrm{sin}\:\theta{d}\theta \\…

let-A-0-dx-x-2-3-x-4-e-i-with-0-lt-lt-pi-2-1-calculate-A-interms-of-2-determine-also-0-dx-x-2-3-x-4-e-i-2-

Question Number 67932 by mathmax by abdo last updated on 02/Sep/19 $${let}\:{A}\left(\theta\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)\left({x}^{\mathrm{4}} −{e}^{{i}\theta} \right)}\:\:{with}\:\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}\left(\theta\right)\:{interms}\:{of}\:\theta \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{also}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)\left({x}^{\mathrm{4}} −{e}^{{i}\theta}…

Question-133452

Question Number 133452 by bagjagunawan last updated on 22/Feb/21 Answered by mnjuly1970 last updated on 22/Feb/21 $$\boldsymbol{\phi}\overset{{x}=\mathrm{2}{y}−\mathrm{1}} {=}\int_{\frac{\mathrm{1}}{\mathrm{2}}\:} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{2}−\mathrm{2}{y}\right){ln}\left({ln}\left(\mathrm{2}{y}\right)\right)}{\mathrm{2}{y}}\left(\mathrm{2}\right){dy} \\ $$$$=\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\:\mathrm{1}} \left({ln}\left(\mathrm{2}\right)+{ln}\left(\mathrm{1}−{y}\right)\right)\left({ln}\left(\mathrm{2}\right)+{ln}\left({y}\right)\right)\frac{{dy}}{{y}} \\…

Question-67907

Question Number 67907 by A8;15: last updated on 02/Sep/19 Commented by mathmax by abdo last updated on 02/Sep/19 $${let}\:{I}\:=\int\:\frac{{dx}}{{x}\sqrt{{x}^{\mathrm{2}} +{x}−\mathrm{6}}} \\ $$$${x}^{\mathrm{2}} +{x}−\mathrm{6}=\mathrm{0}\rightarrow\Delta=\mathrm{1}−\mathrm{4}\left(−\mathrm{6}\right)\:=\mathrm{25}\:\Rightarrow{x}_{\mathrm{1}} =\frac{−\mathrm{1}+\mathrm{5}}{\mathrm{2}}=\mathrm{2}\:\:{and}\: \\…

Question-133440

Question Number 133440 by bagjagunawan last updated on 22/Feb/21 Answered by Ar Brandon last updated on 22/Feb/21 $$\mathrm{f}\left({x}\right)={x}+\mathrm{5}+\sqrt{\mathrm{8}{x}}+\sqrt{\mathrm{12}{x}}+\sqrt{\mathrm{24}}\: \\ $$$$\:\:\:\:\:\:\:\:\:={x}+\left(\sqrt{\mathrm{8}}+\sqrt{\mathrm{12}}\right)\sqrt{{x}}+\mathrm{5}+\sqrt{\mathrm{24}}=\left(\sqrt{{x}}+\sqrt{\mathrm{2}}+\sqrt{\mathrm{3}}\right)^{\mathrm{2}} \\ $$$$\mathcal{I}=\int\mathrm{f}\left({x}\right)\mathrm{d}{x}=\int\sqrt{\left(\sqrt{{x}}+\sqrt{\mathrm{2}}+\sqrt{\mathrm{3}}\right)^{\mathrm{2}} }\mathrm{d}{x} \\ $$$$\:\:\:=\int\left(\sqrt{{x}}+\sqrt{\mathrm{2}}+\sqrt{\mathrm{3}}\right)\mathrm{d}{x}=\frac{\mathrm{2}}{\mathrm{3}}\sqrt{{x}^{\mathrm{3}}…

hi-everybody-with-I-0-1-x-4-1-dx-prove-that-2I-0-x-2-1-x-4-1-dx-

Question Number 133433 by greg_ed last updated on 22/Feb/21 $$\boldsymbol{\mathrm{hi}},\:\boldsymbol{\mathrm{everybody}}\:! \\ $$$$\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{I}}=\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{\boldsymbol{{x}}^{\mathrm{4}} +\mathrm{1}}\:\boldsymbol{{dx}}, \\ $$$$\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\::\:\mathrm{2}\boldsymbol{\mathrm{I}}=\int_{\mathrm{0}} ^{\infty} \:\frac{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}}{\boldsymbol{{x}}^{\mathrm{4}} +\mathrm{1}}\:\boldsymbol{{dx}}. \\ $$ Answered by…

x-dx-

Question Number 133426 by metamorfose last updated on 22/Feb/21 $$\int\lfloor{x}\rfloor{dx}=?… \\ $$ Answered by MJS_new last updated on 22/Feb/21 $$\mathrm{for}\:{a}<{b}:\:\underset{{a}} {\overset{{b}} {\int}}\lfloor{x}\rfloor{dx}=\lfloor{a}\rfloor\underset{{a}} {\overset{\lceil{a}\rceil} {\int}}{dx}+\underset{\lceil{a}\rceil} {\overset{\lfloor{b}\rfloor−\mathrm{1}}…