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Category: Integration

let-f-t-0-e-ax-e-bx-x-2-e-tx-2-dx-with-t-gt-0-1-calculate-f-t-2-find-a-simple-form-of-f-t-3-find-the-value-of-0-e-2x-e-x-x-2-e-3x-2-dx-

Question Number 35821 by prof Abdo imad last updated on 24/May/18 $${let}\:{f}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{ax}} \:−{e}^{−{bx}} }{{x}^{\mathrm{2}} }\:{e}^{−{tx}^{\mathrm{2}} } {dx}\:\:\:{with}\:{t}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{'} \left({t}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({t}\right) \\…

solve-in-R-i-x-x-3x-ii-x-2-3-x-2-0-

Question Number 166892 by mnjuly1970 last updated on 01/Mar/22 $$ \\ $$$$\:\:\:{solve}\:{in}\:\:\mathbb{R} \\ $$$$ \\ $$$$\:\:{i}:\:\:\:\:\lfloor\:{x}\:\lfloor\:{x}\rfloor\rfloor=\:\mathrm{3}{x} \\ $$$$ \\ $$$$\:\:{ii}\::\:\:\:\lfloor{x}\:\rfloor^{\:\mathrm{2}} −\mathrm{3}\:\lfloor{x}\:\rfloor\:+\mathrm{2}\:\leqslant\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:−−−−−− \\ $$$$…

1-sec-4-x-tan-x-sec-4-x-4-dx-2-x-2x-2lnx-2-dx-3-0-1-1-x-2-dx-

Question Number 101345 by bobhans last updated on 02/Jul/20 $$\left(\mathrm{1}\right)\int\:\frac{\mathrm{sec}\:^{\mathrm{4}} {x}\:\mathrm{tan}\:{x}}{\mathrm{sec}\:^{\mathrm{4}} {x}+\mathrm{4}}\:{dx}= \\ $$$$\left(\mathrm{2}\right)\:\int{x}^{\mathrm{2}{x}} \left(\mathrm{2ln}{x}\:+\mathrm{2}\right)\:{dx}\:= \\ $$$$\left(\mathrm{3}\right)\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\:{dx}\:=\: \\ $$ Commented by john…

this-i-a-beautifull-old-question-in-the-forum-by-sir-Ali-Esam-i-Reposted-it-trying-to-find-any-idea-to-solve-I-1-1-sin-x-sinh-1-x-sin-1-x-sinh-x-dx-i-solved-it-numeri

Question Number 101328 by  M±th+et+s last updated on 01/Jul/20 $${this}\:{i}\:{a}\:{beautifull}\:{old}\:{question}\:{in}\:{the}\:{forum} \\ $$$${by}\:{sir}.{Ali}\:{Esam}\:{i}\:{Reposted}\:{it}\:{trying}\:{to} \\ $$$${find}\:{any}\:{idea}\:{to}\:{solve} \\ $$$$ \\ $$$${I}=\int_{−\mathrm{1}} ^{\mathrm{1}} \left(\frac{{sin}\left({x}\right)}{{sinh}^{−\mathrm{1}} \left({x}\right)}\right)\left(\frac{{sin}^{−\mathrm{1}} \left({x}\right)}{{sinh}\left({x}\right)}\right){dx} \\ $$$$ \\…

calculate-If-f-x-x-2-1-x-2-x-2-x-4-1-x-2-2x-3-16-1-3-x-2-3-1-x-x-2-then-f-1-m-n-

Question Number 166829 by mnjuly1970 last updated on 01/Mar/22 $$ \\ $$$$\:\:\:\:\:\:\:\:{calculate}\: \\ $$$$\:\:\:\mathrm{I}{f}\:,\:\:\:\:{f}\left({x}\right)=\frac{\:\left({x}^{\mathrm{2}} +\mathrm{1}\right)\sqrt[{\mathrm{3}}]{\left({x}^{\:\mathrm{2}} +{x}−\mathrm{2}\right)\left({x}^{\:\mathrm{4}} −\mathrm{1}\right)\left({x}^{\:\mathrm{2}} +\mathrm{2}{x}−\mathrm{3}\right)+\mathrm{16}}\:\:+\:\sqrt{{x}^{\:\mathrm{2}} +\mathrm{3}}}{\left(\:\mathrm{1}+{x}\:+{x}^{\:\mathrm{2}} \right)} \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{then}\:,\:\:\:\:{f}\:'\:\left(\mathrm{1}\:\right)\:=?\:\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n} \\…

x-m-x-n-2-x-dx-

Question Number 101286 by student work last updated on 01/Jul/20 $$\int\frac{\left(\mathrm{x}^{\mathrm{m}} −\mathrm{x}^{\mathrm{n}} \right)^{\mathrm{2}} }{\:\sqrt{\mathrm{x}}}\mathrm{dx}=? \\ $$ Commented by student work last updated on 01/Jul/20 $$\mathrm{I}\:\mathrm{need}\:\mathrm{u}\:\mathrm{help}…

x-m-x-n-x-dx-

Question Number 101285 by student work last updated on 01/Jul/20 $$\int\frac{\left(\mathrm{x}^{\mathrm{m}} −\mathrm{x}^{\mathrm{n}} \right)}{\:\sqrt{\mathrm{x}}}\mathrm{dx}=? \\ $$ Commented by bobhans last updated on 01/Jul/20 $$\int\:\left({x}^{{m}−\frac{\mathrm{1}}{\mathrm{2}}} \:−\:{x}^{{n}−\frac{\mathrm{1}}{\mathrm{2}}} \right)\:{dx}\:=\:\frac{\mathrm{2}{x}^{{m}+\frac{\mathrm{1}}{\mathrm{2}}}…

sec-x-dx-

Question Number 101272 by bemath last updated on 01/Jul/20 $$\int\sqrt{\mathrm{sec}\:{x}}\:{dx}\: \\ $$ Commented by john santu last updated on 01/Jul/20 $$\mathrm{I}=\:\int\frac{{dx}}{\:\sqrt{\mathrm{cos}\:{x}}}\:=\:\int\:\frac{\mathrm{1}+\mathrm{sin}\:{x}−\mathrm{sin}\:{x}}{\:\sqrt{\mathrm{cos}\:{x}}}\:{dx} \\ $$$$\mathrm{I}_{\mathrm{1}} \:=\:\int\:\frac{\left(\mathrm{cos}\:\frac{{x}}{\mathrm{2}}−\mathrm{sin}\:\frac{{x}}{\mathrm{2}}\right)^{\mathrm{2}} \:{dx}}{\:\sqrt{\mathrm{cos}\:^{\mathrm{2}}…

find-xdx-x-2-x-1-x-2-x-1-

Question Number 101271 by mathmax by abdo last updated on 01/Jul/20 $$\mathrm{find}\:\int\:\:\:\frac{\mathrm{xdx}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}+\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}}} \\ $$ Commented by Dwaipayan Shikari last updated on 01/Jul/20 $$\int\frac{{x}\left(\sqrt{{x}^{\mathrm{2}}…