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Category: Integration

0-0-0-0-dx-dy-dz-dt-cosh-x-cosh-y-cosh-z-cosh-t-4-7-3-6-12-

Question Number 125919 by Eric002 last updated on 15/Dec/20 $$\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \frac{{dx}\:{dy}\:{dz}\:{dt}}{\left({cosh}\left({x}\right)+{cosh}\left({y}\right)+{cosh}\left({z}\right)+{cosh}\left({t}\right)\right)^{\mathrm{4}} } \\ $$$$=\frac{\mathrm{7}\zeta\left(\mathrm{3}\right)−\mathrm{6}}{\mathrm{12}} \\ $$ Terms of…

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

Question Number 125897 by john_santu last updated on 15/Dec/20 $$\:\:\:\int\:\frac{{dx}}{\mathrm{2}+{x}+\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:?\: \\ $$ Commented by MJS_new last updated on 15/Dec/20 $$\mathrm{the}\:\mathrm{path}\:\mathrm{is} \\ $$$${t}=\frac{\mathrm{1}+\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{{x}}\:\Leftrightarrow\:{x}=\frac{\mathrm{2}{t}}{{t}^{\mathrm{2}} +\mathrm{1}}\:\rightarrow\:{dx}=−\frac{{x}^{\mathrm{2}}…

dx-tan-4-x-1-

Question Number 125895 by john_santu last updated on 15/Dec/20 $$\:\:\:\:\int\:\frac{{dx}}{\mathrm{tan}\:^{\mathrm{4}} {x}+\mathrm{1}}\:? \\ $$ Answered by Dwaipayan Shikari last updated on 15/Dec/20 $$\int\frac{{dx}}{{tan}^{\mathrm{4}} {x}+\mathrm{1}}\:\:\:\:\:\:\:\:\:{tanx}={t}\Rightarrow{sec}^{\mathrm{2}} {x}=\frac{{dt}}{{dx}} \\…

tan-3-x-sec-x-dx-

Question Number 125889 by bramlexs22 last updated on 15/Dec/20 $$\:\:\int\:\frac{\mathrm{tan}\:^{\mathrm{3}} {x}}{\:\sqrt{\mathrm{sec}\:{x}}}\:{dx}\:?\: \\ $$ Answered by bobhans last updated on 15/Dec/20 $$\int\:\frac{\mathrm{tan}\:{x}\left(\mathrm{sec}\:^{\mathrm{2}} {x}−\mathrm{1}\right)}{\:\sqrt{\mathrm{sec}\:{x}}}\:{dx}\:= \\ $$$$\:\left[\:\mathrm{sec}\:{x}\:=\:{u}^{\mathrm{2}} \:\Rightarrow\mathrm{tan}\:{x}\:{dx}\:=\:\frac{\mathrm{2}\:{du}}{{u}}\:\right]…