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Category: Integration

Question-162117

Question Number 162117 by CM last updated on 27/Dec/21 Commented by cortano last updated on 27/Dec/21 $$\:\int\:{x}^{\mathrm{2}} \:\sqrt{\mathrm{4}\:−\left(\mathrm{1}−{x}\right)^{\mathrm{2}} }\:{dx}\:=? \\ $$$$\:\left[\:\mathrm{1}−{x}\:=\:\mathrm{2sin}\:{t}\:\Rightarrow{dx}=−\mathrm{2cos}\:{t}\:{dt}\:\right] \\ $$$$\:\int\:\left(\mathrm{1}−\mathrm{2sin}\:{t}\right)^{\mathrm{2}} \:\left(−\mathrm{2cos}\:^{\mathrm{2}} {t}\right){dt}…

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

Question Number 162112 by amin96 last updated on 26/Dec/21 $$\int\frac{\:\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)}{\left(\mathrm{1}−\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)\right)^{\mathrm{2}} }\boldsymbol{{dx}}=? \\ $$ Answered by Ar Brandon last updated on 26/Dec/21 $${I}=\int\frac{\mathrm{cos}{x}}{\left(\mathrm{1}−\mathrm{cos}{x}\right)^{\mathrm{2}} }{dx}=\int\frac{\mathrm{cos}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}−\mathrm{sin}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{\mathrm{4sin}^{\mathrm{4}}…