Question Number 200606 by Calculusboy last updated on 20/Nov/23 Answered by Frix last updated on 20/Nov/23 $$\int\frac{\mathrm{cos}\:{x}\:\mathrm{sin}^{\mathrm{2}} \:{x}}{\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}}{dx}\:\overset{{t}={x}−\frac{\pi}{\mathrm{4}}} {=} \\ $$$$=\int\left(\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{cos}\:{t}\:\mathrm{sin}\:{t}}{\mathrm{2}}−\frac{\mathrm{sin}^{\mathrm{2}} \:{t}}{\mathrm{2}}−\frac{\mathrm{tan}\:{t}}{\mathrm{4}}\right){dt}= \\ $$$$=\frac{{t}}{\mathrm{4}}−\frac{\mathrm{cos}^{\mathrm{2}} \:{t}}{\mathrm{4}}−\frac{{t}+\mathrm{cos}\:{t}\:\mathrm{sin}\:{t}}{\mathrm{4}}+\frac{\mathrm{ln}\:\mathrm{cos}\:{t}}{\mathrm{4}}=…
Question Number 200601 by Calculusboy last updated on 20/Nov/23 Commented by Frix last updated on 21/Nov/23 $$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{think}\:\mathrm{an}\:\mathrm{exact}\:\mathrm{solution}\:\mathrm{is}\:\mathrm{possible}. \\ $$ Commented by York12 last updated on…
Question Number 200602 by Calculusboy last updated on 20/Nov/23 Answered by Frix last updated on 21/Nov/23 $$\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}} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{n}} \left(\mathrm{n}−\mathrm{x}\right)^{\mathrm{p}+\mathrm{2}} \mathrm{dx}−\mathrm{2n}\int_{\mathrm{0}} ^{\mathrm{n}}…
Question Number 200596 by Rupesh123 last updated on 20/Nov/23 Answered by witcher3 last updated on 22/Nov/23 $$\mathrm{a}\:\mathrm{True}\: \\ $$$$\mathrm{b}\:\mathrm{we}\:\mathrm{can}\:\mathrm{show}\:\mathrm{that}\:\mathrm{exist}\:\mathrm{bijection}\:\mathrm{between}\:\mathrm{som}\:\:\mathrm{set}\:\mathrm{of}\:\mathrm{not}\:\mathrm{differentiabl} \\ $$$$\mathrm{point}\:\mathrm{and}\:\mathbb{N}\: \\ $$$$ \\ $$…
Question Number 200594 by cherokeesay last updated on 20/Nov/23 Answered by Frix last updated on 21/Nov/23 $$\mathrm{Obviously}\:{x}=\mathrm{3}\:\mathrm{is}\:\mathrm{a}\:\mathrm{solution}. \\ $$$$\mathrm{Then}\:\mathrm{you}\:\mathrm{must}\:\mathrm{approximate}…\:\mathrm{I}\:\mathrm{found} \\ $$$${x}\approx.\mathrm{211793616} \\ $$$${x}\approx−\mathrm{2}.\mathrm{80610974} \\ $$…
Question Number 200589 by Ikbal last updated on 20/Nov/23 Answered by Frix last updated on 20/Nov/23 $$\mathrm{First}\:\mathrm{try}\:\mathrm{factors}\:\mathrm{of}\:\pm\mathrm{40}\:\Rightarrow \\ $$$${x}^{\mathrm{4}} −\mathrm{3}{x}^{\mathrm{2}} −\mathrm{42}{x}−\mathrm{40}= \\ $$$$=\left({x}+\mathrm{1}\right)\left({x}−\mathrm{4}\right)\left({x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{10}\right) \\…
Question Number 200586 by Calculusboy last updated on 20/Nov/23 Answered by Frix last updated on 21/Nov/23 $$\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 200579 by sonukgindia last updated on 20/Nov/23 Commented by mr W last updated on 20/Nov/23 $$\Rightarrow{Q}\mathrm{200417} \\ $$ Terms of Service Privacy Policy…
Question Number 200498 by mr W last updated on 19/Nov/23 $${solve}\:{for}\:{x}\in{R} \\ $$$$\sqrt{{x}−\frac{\mathrm{1}}{{x}}}+\sqrt{\mathrm{1}−\frac{\mathrm{1}}{{x}}}={x} \\ $$ Commented by BaliramKumar last updated on 19/Nov/23 $$\mathrm{Q}.\:\mathrm{192958} \\ $$…