Question Number 220730 by Nicholas666 last updated on 18/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{1}}{\:\sqrt{{x}\left(\mathrm{1}\:−\:{x}\right)\left(\mathrm{1}\:+\:{kx}\right)}}\:{dx}\:,\:\left(−\mathrm{1}\:<\:{k}\:<\:\mathrm{1}\right)\:\:\: \\ $$$$ \\ $$ Answered by SdC355 last updated on 18/May/25…
Question Number 220715 by Noorzai last updated on 18/May/25 Commented by Noorzai last updated on 18/May/25 $${do}\:{you}\:{help}\:{me} \\ $$$$ \\ $$ Answered by SdC355 last…
Question Number 220770 by mnjuly1970 last updated on 18/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\:{sin}\left({x}\right)}{\:\sqrt{\mathrm{1}\:+\sqrt{{sin}\left(\mathrm{2}{x}\right)}}}{dx}\: \\ $$ Answered by Ghisom last updated on 21/May/25 $${F}\left({x}\right)=\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}}…
Question Number 220707 by Frix last updated on 18/May/25 $$\underset{−\infty} {\overset{+\infty} {\int}}\frac{{x}\left(\mathrm{tan}^{−\mathrm{1}} \:{x}\right)^{\mathrm{3}} }{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{1}+\mathrm{e}^{\mathrm{4tan}^{−\mathrm{1}} \:{x}} \right)}{dx}=? \\ $$ Answered by SdC355 last updated…
Question Number 220676 by fantastic last updated on 17/May/25 $$\int\:\frac{{xdx}}{\left(\mathrm{1}−{cosx}\right)^{\mathrm{2}} } \\ $$ Answered by Frix last updated on 17/May/25 $${I}=\int\frac{{x}}{\left(\mathrm{1}−\mathrm{cos}\:{x}\right)^{\mathrm{2}} }{dx}\:\overset{\left[\mathrm{by}\:\mathrm{parts}\right]} {=} \\ $$$$=−\frac{\left(\mathrm{2}−\mathrm{cos}\:{x}\right)\left(\mathrm{1}+\mathrm{cos}\:{x}\right)^{\mathrm{2}}…
Question Number 220644 by Nicholas666 last updated on 17/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{1}} ^{\:\mathrm{2}} \:\frac{\mathrm{2}{x}^{\mathrm{2}} }{\:\sqrt{\left(\mathrm{2}{x}\:−\:\mathrm{1}\right)\centerdot\left(\mathrm{2}{x}\:+\:\mathrm{2}\right)}}\:{dx} \\ $$$$ \\ $$ Answered by Ghisom last updated on…
Question Number 220677 by fantastic last updated on 17/May/25 $$\int\sqrt{{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}\:}}{dx} \\ $$ Answered by Frix last updated on 17/May/25 $$\int\sqrt{{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}{dx}\:\overset{\left[{t}=\sqrt{{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}\right]} {=} \\…
Question Number 220674 by Nicholas666 last updated on 17/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\int\int\int_{\:{E}\:} \:\:\frac{{z}^{\mathrm{2}} }{\:\sqrt{{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} }}\:\:{dV} \\ $$$$\:\:\:\:\:\mathrm{with}\:\mathrm{the}\:\mathrm{boundaries}\:\mathrm{of}\:\mathrm{the}\:\mathrm{integration}\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\mathrm{region}\:{E}\:\mathrm{defined}\:\mathrm{by};\: \\ $$$$\:\:\:\:\:\:\bullet\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} +\:{z}^{\mathrm{2}} \:\leqslant\:\mathrm{4}…
Question Number 220544 by Nicholas666 last updated on 15/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{Prove}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{inequality}}; \\ $$$$\:\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{1}\:+\:{x}^{\mathrm{2}} \right)}{\mathrm{1}\:+\:{x}^{\mathrm{2}} }\:{dx}\:<\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{x}\:\mathrm{ln}\left(\mathrm{1}\:+\:{x}^{\mathrm{2}} \right)}{\mathrm{1}\:+\:{x}^{\mathrm{2}} \:}\:{dx}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\:\:\:\:\:\:\:\: \\ $$$$ \\…
Question Number 220526 by Larry last updated on 14/May/25 Answered by Frix last updated on 14/May/25 $$=\int{x}^{−\frac{\mathrm{2}}{\mathrm{7}}} {dx}=\frac{\mathrm{7}}{\mathrm{8}}{x}^{\frac{\mathrm{5}}{\mathrm{7}}} +{C} \\ $$ Answered by SdC355 last…