Question Number 204409 by mr W last updated on 16/Feb/24 $${find}\:\lfloor\int_{\mathrm{0}} ^{\mathrm{2023}} \frac{\mathrm{2}}{{x}+{e}^{{x}} }{dx}\rfloor=? \\ $$ Commented by witcher3 last updated on 16/Feb/24 $$\mathrm{nice}\:\mathrm{problems}\:\mathrm{sir}\:\:\mathrm{Thanx}\:\mathrm{for}\:\mathrm{share}\:\mathrm{it} \\…
Question Number 204275 by Frix last updated on 11/Feb/24 $$\mathrm{Show}\:\mathrm{that} \\ $$$$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{4}}} {\int}}\sqrt{\mathrm{tan}\:{x}}\:\sqrt{\mathrm{1}−\mathrm{tan}\:{x}}\:{dx}=\left(\frac{\sqrt{\sqrt{\mathrm{2}}−\mathrm{1}}}{\:\sqrt{\mathrm{2}}}−\mathrm{1}\right)\pi \\ $$ Answered by witcher3 last updated on 11/Feb/24 $$\mathrm{nice}\:\mathrm{problem} \\…
Question Number 204233 by Perelman last updated on 09/Feb/24 Answered by Frix last updated on 09/Feb/24 $$\int\frac{{x}^{\frac{\mathrm{1}}{\mathrm{2}}} }{{x}^{\frac{\mathrm{3}}{\mathrm{4}}} +\mathrm{1}}{dx}\:\overset{{t}={x}^{\frac{\mathrm{1}}{\mathrm{4}}} } {=}\:\mathrm{4}\int\frac{{t}^{\mathrm{5}} }{{t}^{\mathrm{3}} +\mathrm{1}}{dt}= \\ $$$$=\mathrm{4}\int{t}^{\mathrm{2}}…
Question Number 204244 by pticantor last updated on 09/Feb/24 Answered by pticantor last updated on 09/Feb/24 $${please}\:{can}\:{some}\:{one}\:{help}\:{me}\:{to}\:{solve}\:{question}\:\mathrm{2}−{b}\: \\ $$$${for}\:{this}\:{exercise}? \\ $$ Answered by witcher3 last…
Question Number 203964 by mnjuly1970 last updated on 03/Feb/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{Advanced}\:\:{calculus}\:… \\ $$$$\:\:{Q}:\:\:\:{If}\:,\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{x}\mathrm{ln}\left({x}\right)\mathrm{ln}^{\mathrm{2}} \left({y}\:\right)}{\mathrm{1}−{xy}}\:{dxdy}\:=\:\lambda\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\mathrm{1}}{{n}^{\:\mathrm{3}} \left(\:{n}+\mathrm{1}\:\right)^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:{Find}\:,\:\:\:\:\:\lambda\:=?…
Question Number 203884 by Mathspace last updated on 31/Jan/24 $${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{3}} }{\left(\mathrm{1}+{x}\right)^{\mathrm{4}} \left({x}+\mathrm{2}\right)^{\mathrm{5}} }{dx} \\ $$ Answered by Frix last updated on 01/Feb/24 $$\mathrm{Use}\:\mathrm{Ostrogradski}'\mathrm{s}\:\mathrm{Method}\:\mathrm{to}\:\mathrm{get}…
Question Number 203867 by Mathspace last updated on 30/Jan/24 $${find}\:\int\sqrt{\frac{\mathrm{1}−{x}^{\mathrm{3}} }{\mathrm{1}+{x}^{\mathrm{3}} }}{dx} \\ $$ Answered by MathematicalUser2357 last updated on 06/Feb/24 $$\frac{\mathrm{8}{x}\sqrt{\frac{\mathrm{1}−{x}^{\mathrm{3}} }{{x}^{\mathrm{3}} +\mathrm{1}}}{F}_{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{3}};−\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}};\frac{\mathrm{4}}{\mathrm{3}};{x}^{\mathrm{3}}…
Question Number 203836 by patrice last updated on 29/Jan/24 Answered by Frix last updated on 30/Jan/24 $$\mathrm{In}\:\mathrm{3}\:\mathrm{steps}: \\ $$$$\mathrm{1}.\:{t}={x}^{\mathrm{2}} −\mathrm{1} \\ $$$$\mathrm{2}.\:{u}=\frac{\mathrm{5}{t}+\sqrt{\mathrm{25}{t}^{\mathrm{2}} +\mathrm{11}}}{\:\mathrm{11}} \\ $$$$\mathrm{3}.\:{v}=\frac{\sqrt{\mathrm{1}+{u}}}{\:\sqrt{\mathrm{1}−{u}}}\:\Rightarrow…
Question Number 203772 by Calculusboy last updated on 27/Jan/24 Answered by witcher3 last updated on 27/Jan/24 $$\mathrm{no}\:\mathrm{close}\:\mathrm{formes}\:\mathrm{just}\:\mathrm{series}\:\mathrm{or}\:\mathrm{aproximination} \\ $$ Commented by Calculusboy last updated on…
Question Number 203747 by patrice last updated on 27/Jan/24 Answered by esmaeil last updated on 27/Jan/24 $${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{x}}{\mathrm{1}+{cosx}}{dx}+\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{sinx}}{\mathrm{1}+{cosx}}{dx} \\ $$$${x}={u}\rightarrow{dx}={du} \\ $$$$\frac{{dx}}{\mathrm{1}+{cosx}}={dv}\rightarrow{v}={tan}\frac{{x}}{\mathrm{2}}…