Question Number 99168 by bemath last updated on 19/Jun/20 Answered by Kunal12588 last updated on 19/Jun/20 $${I}=\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \frac{{x}\mathrm{cos}\:{x}}{\mathrm{1}+\mathrm{sin}^{\mathrm{2}} \:{x}}{dx} \\ $$$$\Rightarrow{I}=\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \:\frac{\left(\pi/\mathrm{2}\right)\mathrm{cos}\:{x}}{\mathrm{1}+\mathrm{sin}^{\mathrm{2}} \:{x}}{dx}−\int_{\mathrm{0}}…
Question Number 33619 by NECx last updated on 20/Apr/18 $$\int{x}^{\mathrm{5}/\mathrm{2}} \left(\mathrm{1}−{x}\right)^{\mathrm{3}/\mathrm{2}} {dx} \\ $$ Commented by abdo imad last updated on 20/Apr/18 $${let}\:{put}\:{I}\:=\:\int\:\:{x}^{\frac{\mathrm{5}}{\mathrm{2}}} \:\left(\mathrm{1}−{x}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} \:{dx}\:\:.{ch}.{x}={sin}^{\mathrm{2}}…
Question Number 99154 by bramlex last updated on 19/Jun/20 Answered by bramlex last updated on 19/Jun/20 Answered by abdomathmax last updated on 19/Jun/20 $$\left.\mathrm{a}\right)\:\mathrm{divergent}\:\mathrm{integral} \\…
Question Number 164685 by Zaynal last updated on 20/Jan/22 $$\int\mathrm{4}\boldsymbol{{x}}^{\mathrm{2}} \:\left(\mathrm{1}−\boldsymbol{{x}}\right)^{\mathrm{3}} \:\boldsymbol{{dx}} \\ $$ Answered by Ar Brandon last updated on 20/Jan/22 $$=\mathrm{4}\int\left(\mathrm{1}−{x}−\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{1}−{x}\right)^{\mathrm{3}} {dx}…
Question Number 164686 by Zaynal last updated on 20/Jan/22 $$\int\mathrm{36}\boldsymbol{{x}}^{\mathrm{2}} \:\left(\mathrm{2}\boldsymbol{{x}}+\mathrm{3}\right)^{−\mathrm{7}} \:\boldsymbol{{dx}} \\ $$$$\: \\ $$ Answered by Ar Brandon last updated on 20/Jan/22 $$=\mathrm{36}\int\frac{{x}^{\mathrm{2}}…
Question Number 164684 by Zaynal last updated on 20/Jan/22 $$\int\mathrm{24}\left(\mathrm{2}\boldsymbol{{x}}−\mathrm{1}\right)^{−\mathrm{3}} \:\boldsymbol{{dx}} \\ $$ Answered by Ar Brandon last updated on 20/Jan/22 $$=\int\frac{\mathrm{24}}{\left(\mathrm{2}{x}−\mathrm{1}\right)^{\mathrm{3}} }{dx}=−\frac{\mathrm{6}}{\left(\mathrm{2}{x}−\mathrm{1}\right)^{\mathrm{2}} }+{C} \\…
Question Number 99146 by mathmax by abdo last updated on 18/Jun/20 $$\left.\mathrm{1}\right)\:\mathrm{explicit}\:\mathrm{f}\left(\mathrm{a}\right)\:=\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \:\:\mathrm{arctan}\left(\frac{\mathrm{a}}{\mathrm{x}}\right)\mathrm{dx}\:\:\mathrm{with}\:\mathrm{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{calculate}\:\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \:\mathrm{arctan}\left(\frac{\mathrm{2}}{\mathrm{x}}\right)\mathrm{dx}\:\mathrm{and}\:\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \:\:\mathrm{arctan}\left(\frac{\mathrm{3}}{\mathrm{x}}\right)\mathrm{dx} \\ $$ Answered by mathmax…
Question Number 164681 by Zaynal last updated on 20/Jan/22 $$\int\:\left(−\mathrm{8}\boldsymbol{{x}}\sqrt{\mathrm{3}−\mathrm{2}\boldsymbol{{x}}}\:\right)\boldsymbol{{dx}} \\ $$ Answered by Ar Brandon last updated on 20/Jan/22 $$=\mathrm{4}\int\left(\mathrm{3}−\mathrm{2}{x}−\mathrm{3}\right)\sqrt{\mathrm{3}−\mathrm{2}{x}}{dx} \\ $$$$=\mathrm{4}\int\left(\left(\mathrm{3}−\mathrm{2}{x}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} −\mathrm{3}\left(\mathrm{3}−\mathrm{2}{x}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \right){dx}…
Question Number 164683 by Zaynal last updated on 20/Jan/22 $$\int\frac{\boldsymbol{{x}}^{\mathrm{2}} }{\left(\boldsymbol{{x}}+\mathrm{7}\right)^{\mathrm{4}} }\boldsymbol{{dx}} \\ $$ Answered by Ar Brandon last updated on 20/Jan/22 $$=\int\frac{\left({x}+\mathrm{7}−\mathrm{7}\right)^{\mathrm{2}} }{\left({x}+\mathrm{7}\right)^{\mathrm{4}} }{dx}…
Question Number 164682 by Zaynal last updated on 20/Jan/22 $$\int\mathrm{27}\boldsymbol{{x}}^{\mathrm{2}} \:\sqrt{\mathrm{3}\boldsymbol{{x}}−\mathrm{2}}\:\boldsymbol{{dx}} \\ $$ Answered by Ar Brandon last updated on 20/Jan/22 $$=\mathrm{3}\int\left(\mathrm{3}{x}−\mathrm{2}+\mathrm{2}\right)^{\mathrm{2}} \sqrt{\mathrm{3}{x}−\mathrm{2}}{dx} \\ $$$$=\mathrm{3}\int\left(\left(\mathrm{3}{x}−\mathrm{2}\right)^{\frac{\mathrm{5}}{\mathrm{2}}}…