Category: Integration

Question Number 94084 by seedhamaieng@gmail.com last updated on 16/May/20 $$\int\sqrt{\mathrm{cot}{x}}{dx}\: \\ $$ Commented by Kunal12588 last updated on 16/May/20 $$=\int\sqrt{{tan}\left(\pi/\mathrm{2}−{x}\right)}\:{dx} \\ $$$$=−\int\sqrt{{tan}\:{t}}\:{dt}\:;\:{with}\:{t}={tan}\left(\frac{\pi}{\mathrm{2}}−{x}\right) \\ $$$$\int\sqrt{{tan}\:{x}}\:{dx}\:{is}\:{very}\:{popular}\:{integral},\:{its}\:{already} \\…

Question Number 159612 by cortano last updated on 19/Nov/21 Answered by Ar Brandon last updated on 19/Nov/21 $${I}=\int\frac{\sqrt{\mathrm{2}+\sqrt[{\mathrm{3}}]{{x}}}}{\:\sqrt[{\mathrm{3}}]{{x}}}{dx},\:{x}={u}^{\mathrm{3}} \Rightarrow{dx}=\mathrm{3}{u}^{\mathrm{2}} {du} \\ $$$$\:\:\:=\mathrm{3}\int\frac{\sqrt{\mathrm{2}+{u}}}{{u}}\centerdot{u}^{\mathrm{2}} {du}=\mathrm{3}\int{u}\sqrt{\mathrm{2}+{u}}{du} \\ $$$$\:\:\:=\mathrm{3}\int\left({u}+\mathrm{2}\right)^{\frac{\mathrm{3}}{\mathrm{2}}}…

Question Number 28541 by abdo imad last updated on 26/Jan/18 $${prove}\:{that}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sinx}}{{e}^{{ax}} −\mathrm{1}}{dx}=\:\sum_{{p}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{\mathrm{1}+{p}^{\mathrm{2}} {a}^{\mathrm{2}} }\:\:\:\:\:{with}\:{a}>\mathrm{0} \\ $$ Terms of Service Privacy Policy…

Question Number 159597 by mathlove last updated on 19/Nov/21 Answered by mr W last updated on 19/Nov/21 $$\left(\frac{\mathrm{sin}^{\mathrm{2014}} \:{x}\:\mathrm{sin}\:\left(\mathrm{2014}{x}\right)}{\mathrm{2014}}\right)' \\ $$$$=\frac{\mathrm{2014}\:\mathrm{sin}^{\mathrm{2013}} \:{x}\:\mathrm{cos}\:{x}\:\mathrm{sin}\:\left(\mathrm{2014}{x}\right)+\mathrm{sin}\:^{\mathrm{2014}} {x}×\mathrm{2014}\:\mathrm{cos}\:\left(\mathrm{2014}{x}\right)}{\mathrm{2014}} \\ $$$$=\mathrm{sin}^{\mathrm{2013}}…

Question Number 159552 by Ar Brandon last updated on 18/Nov/21 Commented by Ar Brandon last updated on 18/Nov/21 $$\mathrm{Prove}\:\mathrm{the}\:\mathrm{above}\:\mathrm{results} \\ $$ Commented by mindispower last…