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Category: Integration

dx-cos-x-cosec-x-

Question Number 152291 by iloveisrael last updated on 27/Aug/21 $$\:\int\:\frac{{dx}}{\mathrm{cos}\:{x}+\mathrm{cosec}\:{x}}\:=? \\ $$ Answered by puissant last updated on 27/Aug/21 $${I}=\int\frac{{dx}}{{cosx}+{cosecx}} \\ $$$$=\int\frac{{sinx}}{{cosxsinx}+\mathrm{1}}{dx}\:=\:\int\frac{\mathrm{2}{sinx}}{{sin}\mathrm{2}{x}+\mathrm{2}}{dx} \\ $$$$=\int\frac{{sinx}+{cosx}}{{sin}\mathrm{2}{x}+\mathrm{2}}{dx}+\int\frac{{sinx}−{cosx}}{{sin}\mathrm{2}{x}+\mathrm{2}}{dx} \\…

0-pi-2-sin-2xlog-tan-x-dx-

Question Number 152275 by peter frank last updated on 27/Aug/21 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{sin}\:\mathrm{2xlog}\left(\:\mathrm{tan}\:\mathrm{x}\right)\mathrm{dx} \\ $$ Answered by qaz last updated on 27/Aug/21 $$\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{sin}\:\mathrm{2x}\centerdot\mathrm{lntan}\:\mathrm{xdx}…

tan-tan-3-1-tan-3-d-

Question Number 152273 by peter frank last updated on 27/Aug/21 $$\int\:\frac{\mathrm{tan}\:\theta+\mathrm{tan}\:^{\mathrm{3}} \theta}{\mathrm{1}+\mathrm{tan}\:^{\mathrm{3}} \theta}\mathrm{d}\theta \\ $$ Answered by qaz last updated on 27/Aug/21 $$\int\frac{\mathrm{tan}\:\theta+\mathrm{tan}\:^{\mathrm{3}} \theta}{\mathrm{1}+\mathrm{tan}\:^{\mathrm{3}} \theta}\mathrm{d}\theta…

5x-3-x-2-4x-10-dx-

Question Number 152270 by peter frank last updated on 27/Aug/21 $$\int\frac{\mathrm{5x}+\mathrm{3}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{4x}+\mathrm{10}}}\mathrm{dx} \\ $$ Answered by Olaf_Thorendsen last updated on 27/Aug/21 $$\mathrm{F}\left({x}\right)\:=\:\int\frac{\mathrm{5}{x}+\mathrm{3}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{10}}}\:{dx} \\ $$$$\mathrm{F}\left({u}−\mathrm{2}\right)\:=\:\int\frac{\mathrm{5}{u}−\mathrm{7}}{\:\sqrt{{u}^{\mathrm{2}}…

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

Question Number 152271 by peter frank last updated on 27/Aug/21 $$\int\left(\mathrm{3x}−\mathrm{2}\right)\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}\:\mathrm{dx} \\ $$ Answered by qaz last updated on 27/Aug/21 $$\mathrm{A}=\int\left(\mathrm{3x}−\mathrm{2}\right)\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}\mathrm{dx} \\ $$$$=\frac{\mathrm{3}}{\mathrm{2}}\int\left(\mathrm{2x}+\mathrm{1}\right)\sqrt{\mathrm{x}^{\mathrm{2}}…