Question Number 200601 by Calculusboy last updated on 20/Nov/23 Commented by Frix last updated on 21/Nov/23 $$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{think}\:\mathrm{an}\:\mathrm{exact}\:\mathrm{solution}\:\mathrm{is}\:\mathrm{possible}. \\ $$ Commented by York12 last updated on…
Question Number 200606 by Calculusboy last updated on 20/Nov/23 Answered by Frix last updated on 20/Nov/23 $$\int\frac{\mathrm{cos}\:{x}\:\mathrm{sin}^{\mathrm{2}} \:{x}}{\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}}{dx}\:\overset{{t}={x}−\frac{\pi}{\mathrm{4}}} {=} \\ $$$$=\int\left(\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{cos}\:{t}\:\mathrm{sin}\:{t}}{\mathrm{2}}−\frac{\mathrm{sin}^{\mathrm{2}} \:{t}}{\mathrm{2}}−\frac{\mathrm{tan}\:{t}}{\mathrm{4}}\right){dt}= \\ $$$$=\frac{{t}}{\mathrm{4}}−\frac{\mathrm{cos}^{\mathrm{2}} \:{t}}{\mathrm{4}}−\frac{{t}+\mathrm{cos}\:{t}\:\mathrm{sin}\:{t}}{\mathrm{4}}+\frac{\mathrm{ln}\:\mathrm{cos}\:{t}}{\mathrm{4}}=…
Question Number 200602 by Calculusboy last updated on 20/Nov/23 Answered by Frix last updated on 21/Nov/23 $$\mathrm{Use}\:{u}'=\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cos}\:{t}}\:\rightarrow\:{u}=\mathrm{tan}\:\frac{{t}}{\mathrm{2}}; \\ $$$$\:\:\:\:\:\:\:\:\:{v}={t}+\mathrm{sin}\:{t}\:\rightarrow\:{v}'=\mathrm{1}+\mathrm{cos}\:{t} \\ $$$$\int\frac{{t}+\mathrm{sin}\:{t}}{\mathrm{1}+\mathrm{cos}\:{t}}{dt}= \\ $$$$=\left({t}+\mathrm{sin}\:{t}\right)\mathrm{tan}\:\frac{{t}}{\mathrm{2}}\:−\int\left(\mathrm{1}+\mathrm{cos}\:{t}\right)\:\mathrm{tan}\:\frac{{t}}{\mathrm{2}}\:{dt}= \\ $$$$={t}\mathrm{tan}\:\frac{{t}}{\mathrm{2}}\:+\mathrm{1}−\mathrm{cos}\:{t}\:−\int\mathrm{sin}\:{t}\:{dt}=…
Question Number 200603 by Calculusboy last updated on 20/Nov/23 Answered by witcher3 last updated on 20/Nov/23 $$\mathrm{x}^{\mathrm{2}} =\left(\mathrm{n}−\mathrm{x}\right)^{\mathrm{2}} −\mathrm{2n}\left(\mathrm{n}−\mathrm{x}\right)+\mathrm{n}^{\mathrm{2}} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{n}} \left(\mathrm{n}−\mathrm{x}\right)^{\mathrm{p}+\mathrm{2}} \mathrm{dx}−\mathrm{2n}\int_{\mathrm{0}} ^{\mathrm{n}}…
Question Number 200586 by Calculusboy last updated on 20/Nov/23 Answered by Frix last updated on 21/Nov/23 $$\underset{\mathrm{0}} {\overset{\pi} {\int}}\frac{{x}}{\mathrm{2}+\mathrm{tan}^{\mathrm{2}} \:{x}}{dx}=\underset{\mathrm{0}} {\overset{\pi} {\int}}{xdx}−\underset{\mathrm{0}} {\overset{\pi} {\int}}\frac{{x}}{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \:{x}}…
Question Number 200474 by Rupesh123 last updated on 19/Nov/23 Answered by witcher3 last updated on 19/Nov/23 $$\mathrm{erf}\left(\mathrm{x}\right)=\frac{\mathrm{2}}{\:\sqrt{\pi}}\int_{\mathrm{0}} ^{\mathrm{x}} \mathrm{e}^{−\mathrm{t}^{\mathrm{2}} } \mathrm{dt} \\ $$$$\mathrm{ln}\left(\mathrm{x}+\mathrm{ln}\left(\mathrm{x}\right)\right)=\int_{\mathrm{0}} ^{\mathrm{5}} \mathrm{e}^{−\mathrm{t}^{\mathrm{2}}…
Question Number 200464 by Frix last updated on 19/Nov/23 $$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\frac{{dx}}{\mathrm{1}+\mathrm{tan}^{\mathrm{2023}} \:{x}}=??????? \\ $$ Answered by som(math1967) last updated on 19/Nov/23 $$\:{I}=\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\frac{{cos}^{\mathrm{2023}}…
Question Number 200444 by Calculusboy last updated on 18/Nov/23 Answered by Frix last updated on 19/Nov/23 $$\int\mathrm{e}^{−\mathrm{i}{x}^{\mathrm{2}} } {dx}\:\overset{{t}=\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{4}}} {x}} {=} \\ $$$$=\frac{\sqrt{\mathrm{2}}\left(\mathrm{1}−\mathrm{i}\right)}{\mathrm{2}}\int\mathrm{e}^{−{t}^{\mathrm{2}} } {dt}=\frac{\sqrt{\mathrm{2}\pi}\left(\mathrm{1}−\mathrm{i}\right)}{\mathrm{4}}\int\frac{\mathrm{2e}^{{t}^{\mathrm{3}}…
Question Number 200403 by Anonim_X last updated on 18/Nov/23 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\int\:\:\frac{\left(\boldsymbol{{x}}^{\mathrm{2}} \:+\:\:\mathrm{1}\right)\boldsymbol{{dx}}}{\boldsymbol{{x}}\left(\boldsymbol{{x}}−\mathrm{1}\right)\left(\boldsymbol{{x}}+\mathrm{1}\right)}\:=\:?? \\ $$$$ \\ $$ Answered by cortano12 last updated on 18/Nov/23 $$\:\mathrm{I}\:=\:\int\:\frac{\mathrm{x}^{\mathrm{2}}…
Question Number 200366 by ajfour last updated on 17/Nov/23 Commented by ajfour last updated on 17/Nov/23 $${The}\:{red}\:{circular}\:{arc}\:{length}\:{is}\:{equal} \\ $$$$\:{to}\:{blue}\:{arclength}\:\:{of}\:{parabola}.\:{Find}\: \\ $$$${the}\:{equation}\:{of}\:{parabola}\:{in}\:{the}\:{form}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}={ax}^{\mathrm{2}} +{c}. \\…