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Category: Integration

Question-141533

Question Number 141533 by sarkor last updated on 20/May/21 Answered by qaz last updated on 20/May/21 $$\int\frac{{dx}}{\mathrm{1}+\mathrm{3cos}\:^{\mathrm{2}} {x}}=\int\frac{\mathrm{sin}\:^{\mathrm{2}} {x}+\mathrm{cos}\:^{\mathrm{2}} {x}}{\mathrm{sin}\:^{\mathrm{2}} {x}+\mathrm{4cos}\:^{\mathrm{2}} {x}}{dx} \\ $$$$=\int\frac{\mathrm{tan}\:^{\mathrm{2}} {x}+\mathrm{1}}{\mathrm{tan}\:^{\mathrm{2}}…

Question-141532

Question Number 141532 by sarkor last updated on 20/May/21 Answered by qaz last updated on 20/May/21 $$\int\frac{\mathrm{3sin}\:{x}−\mathrm{2cos}\:{x}}{\mathrm{1}+\mathrm{cos}\:{x}}{dx} \\ $$$$=\int\frac{\mathrm{6sin}\:\frac{{x}}{\mathrm{2}}\mathrm{cos}\:\frac{{x}}{\mathrm{2}}−\mathrm{4cos}\:^{\mathrm{2}} \frac{{x}}{\mathrm{2}}+\mathrm{2}}{\mathrm{2cos}\:^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{dx} \\ $$$$=\int\left(\mathrm{3tan}\:\frac{{x}}{\mathrm{2}}−\mathrm{2}+\mathrm{sec}\:^{\mathrm{2}} \frac{{x}}{\mathrm{2}}\right){dx} \\…

Question-141534

Question Number 141534 by sarkor last updated on 20/May/21 Answered by som(math1967) last updated on 20/May/21 $$\int\boldsymbol{{cos}}^{\mathrm{2}} \boldsymbol{{xsin}}^{\mathrm{8}} \boldsymbol{{xcosxdx}} \\ $$$$=\int\left(\mathrm{1}−\boldsymbol{{sin}}^{\mathrm{2}} \boldsymbol{{x}}\right)\boldsymbol{{sin}}^{\mathrm{8}} \boldsymbol{{xd}}\left(\boldsymbol{{sinx}}\right) \\ $$$$=\int\boldsymbol{{sin}}^{\mathrm{8}}…

Question-141530

Question Number 141530 by sarkor last updated on 20/May/21 Commented by mohammad17 last updated on 20/May/21 $$=\int\:\frac{{x}^{\mathrm{2}} −\mathrm{9}+\mathrm{9}}{\:\sqrt{{x}−\mathrm{3}}}{dx}=\int\:\frac{\left({x}−\mathrm{3}\right)\left({x}+\mathrm{3}\right)}{\:\sqrt{{x}−\mathrm{3}}}{dx}+\int\:\frac{\mathrm{9}}{\:\sqrt{{x}−\mathrm{3}}}{dx} \\ $$$$ \\ $$$$=\int\left(\sqrt{{x}−\mathrm{3}}\right)\left({x}+\mathrm{3}\right){dx}+\int\mathrm{9}\left({x}−\mathrm{3}\right)^{−\frac{\mathrm{1}}{\mathrm{2}}} {dx} \\ $$$$…

Question-141531

Question Number 141531 by sarkor last updated on 20/May/21 Commented by mohammad17 last updated on 20/May/21 $${let}\:{z}^{\mathrm{6}} ={x}+\mathrm{1}\Rightarrow\mathrm{6}{z}^{\mathrm{5}} {dz}={dx} \\ $$$$ \\ $$$$\int\:\:\frac{\mathrm{6}{z}^{\mathrm{3}} \left({z}^{\mathrm{4}} +{z}\right)}{\left({z}+\mathrm{1}\right)}{dz}=\mathrm{6}\int\:\:\frac{{z}^{\mathrm{7}}…

Question-141526

Question Number 141526 by sarkor last updated on 20/May/21 Answered by rs4089 last updated on 20/May/21 $$\int\left({x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}\:}\:{dx} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int\left(\mathrm{2}{x}+\mathrm{2}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}}\:{dx} \\ $$$${let}\:{x}^{\mathrm{2}} +\mathrm{2}{x}={t}\:\Rightarrow\left(\mathrm{2}{x}+\mathrm{2}\right){dx}={dt} \\…