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Category: Integration

Question-227271

Question Number 227271 by Spillover last updated on 11/Jan/26 Answered by Kassista last updated on 11/Jan/26 $$ \\ $$$$\int\:\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} +\mathrm{5}}\:{dx}\:=\:\int\:\frac{{x}^{\mathrm{2}} +\mathrm{5}−\mathrm{5}}{{x}^{\mathrm{2}} +\mathrm{5}}\:{dx}\:=\:\int\:\frac{{x}^{\mathrm{2}} +\mathrm{5}}{{x}^{\mathrm{2}} +\mathrm{5}}\:{dx}\:−\int\frac{\mathrm{5}}{{x}^{\mathrm{2}}…

Question-227128

Question Number 227128 by Spillover last updated on 01/Jan/26 Answered by som(math1967) last updated on 01/Jan/26 $$\int_{\mathrm{0}\:\:} ^{\frac{\pi}{\mathrm{2}}} \frac{\sqrt{{tanx}}}{\mathrm{1}+\sqrt{{tanx}}}{dx} \\ $$$${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\sqrt{{tan}\left(\frac{\pi}{\mathrm{2}}−{x}\right)}{dx}}{\mathrm{1}+\sqrt{{tan}\left(\frac{\pi}{\mathrm{2}}−{x}\right)}} \\ $$$${I}=\int_{\mathrm{0}}…

0-pi-sin-2x-3-cos-x-1-dx-

Question Number 227108 by gregori last updated on 31/Dec/25 $$\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\mathrm{sin}\:\mathrm{2}{x}}{\mathrm{3}}\sqrt{\mathrm{cos}\:{x}+\mathrm{1}}\:{dx}\:=? \\ $$ Answered by Frix last updated on 31/Dec/25 $$=\frac{\mathrm{2}}{\mathrm{3}}\underset{\mathrm{0}} {\overset{\pi} {\int}}\mathrm{cos}\:{x}\:\mathrm{sin}\:{x}\:\sqrt{\mathrm{1}+\mathrm{cos}\:{x}}\:{dx}\:\overset{\left[{t}=\sqrt{\mathrm{1}+\mathrm{cos}\:{x}}\right]} {=}…