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Category: Integration

Question-62613

Question Number 62613 by necx1 last updated on 23/Jun/19 Commented by Prithwish sen last updated on 23/Jun/19 $$=\int\frac{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)\:\mathrm{dx}}{\mathrm{x}^{\mathrm{3}} \sqrt{\frac{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} −\mathrm{x}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} }}}…

sin-100-x-cos-100-x-dx-

Question Number 62596 by aliesam last updated on 23/Jun/19 $$\int\mathrm{sin}^{\mathrm{100}} \left(\mathrm{x}\right)\:\mathrm{cos}^{\mathrm{100}} \left(\mathrm{x}\right)\:\mathrm{dx} \\ $$ Answered by MJS last updated on 23/Jun/19 $$\int\mathrm{sin}^{\mathrm{100}} \:{x}\:\mathrm{cos}^{\mathrm{100}} \:{x}\:{dx}=\int\left(\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}\right)^{\mathrm{100}} {dx}=…

x-2-x-1-x-1-dx-

Question Number 128109 by bemath last updated on 04/Jan/21 $$\emptyset\:=\:\int\:\left(\mathrm{x}−\mathrm{2}\right)\:\sqrt{\frac{\mathrm{x}+\mathrm{1}}{\mathrm{x}−\mathrm{1}}}\:\mathrm{dx}\: \\ $$ Answered by liberty last updated on 04/Jan/21 $$\:\phi\:=\:\int\:\frac{\left(\mathrm{x}−\mathrm{2}\right)\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}}{\mathrm{x}−\mathrm{1}}\:\mathrm{dx}\: \\ $$$$\:\mathrm{let}\:\mathrm{x}\:=\:\mathrm{sec}\:\mathrm{t}\: \\ $$$$\:\emptyset\:=\:\int\:\frac{\left(\mathrm{sec}\:\mathrm{t}−\mathrm{2}\right).\mathrm{tan}\:\mathrm{t}}{\mathrm{sec}\:\mathrm{t}−\mathrm{1}}.\:\left(\mathrm{sec}\:\mathrm{t}\:\mathrm{tan}\:\mathrm{t}\:\right)\:\mathrm{dt}\:…

ln-x-x-2-a-2-dx-

Question Number 128079 by liberty last updated on 04/Jan/21 $$\Omega\:=\:\int\:\mathrm{ln}\:\left(\mathrm{x}+\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{a}^{\mathrm{2}} }\:\right)\mathrm{dx} \\ $$ Answered by bemath last updated on 04/Jan/21 $$\Omega=\mathrm{x}\:\mathrm{ln}\:\left(\mathrm{x}+\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{a}^{\mathrm{2}} }\:\right)−\int\:\frac{\mathrm{x}\left(\mathrm{1}+\frac{\mathrm{x}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{a}^{\mathrm{2}}…

sin-2x-1-x-3-dx-

Question Number 128073 by I want to learn more last updated on 04/Jan/21 $$\int\:\frac{\mathrm{sin}\left(\mathrm{2x}\right)}{\left(\mathrm{1}\:\:−\:\:\mathrm{x}\right)^{\mathrm{3}} }\:\:\mathrm{dx} \\ $$ Answered by mindispower last updated on 04/Jan/21 $$\Leftrightarrow{Im}\int\frac{{e}^{\mathrm{2}{ix}}…

0-1-0-1-1-1-xy-2-dx-dy-

Question Number 128057 by bramlexs22 last updated on 04/Jan/21 $$\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{1}}{\mathrm{1}−\mathrm{xy}^{\mathrm{2}} }\:\mathrm{dx}\:\mathrm{dy}\:=? \\ $$ Answered by liberty last updated on 04/Jan/21 $$\:\frac{\mathrm{1}}{\mathrm{1}−\mathrm{xy}^{\mathrm{2}}…

I-arctan-x-2-x-2-dx-

Question Number 128052 by bramlexs22 last updated on 04/Jan/21 $$\:\mathrm{I}\:=\:\int\:\mathrm{arctan}\:\left(\frac{\mathrm{x}−\mathrm{2}}{\mathrm{x}+\mathrm{2}}\right)\:\mathrm{dx}\: \\ $$ Answered by liberty last updated on 04/Jan/21 $$\mathrm{I}\:=\:\mathrm{x}\:\mathrm{arctan}\:\left(\frac{\mathrm{x}−\mathrm{2}}{\mathrm{x}+\mathrm{2}}\right)−\int\:\mathrm{x}\left(\frac{\mathrm{1}}{\mathrm{1}+\left(\frac{\mathrm{x}−\mathrm{2}}{\mathrm{x}+\mathrm{2}}\right)^{\mathrm{2}} }.\:\frac{\mathrm{4}}{\left(\mathrm{x}+\mathrm{2}\right)^{\mathrm{2}} }\right)\mathrm{dx} \\ $$$$\mathrm{I}=\mathrm{x}\:\mathrm{arctan}\:\left(\frac{\mathrm{x}−\mathrm{2}}{\mathrm{x}+\mathrm{2}}\right)−\int\:\frac{\mathrm{2x}}{\mathrm{x}^{\mathrm{2}} +\mathrm{4}}\:\mathrm{dx}…