Question Number 204279 by liuxinnan last updated on 11/Feb/24 Commented by liuxinnan last updated on 11/Feb/24 $${a}\:{beautiful}\:{image} \\ $$ Terms of Service Privacy Policy Contact:…
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 $$\mathrm{nice}\:\mathrm{problem} \\…
Question Number 204270 by universe last updated on 10/Feb/24 Answered by witcher3 last updated on 10/Feb/24 $$\left(\mathrm{1}\right) \\ $$$$\mathrm{withe}\:\mathrm{recursion}\:\mathrm{We}\:\mathrm{can}\:\mathrm{easly}\:\mathrm{proov}\:\mathrm{that}\:\mathrm{a}_{\mathrm{n}} >\mathrm{0} \\ $$$$\mathrm{a}_{\mathrm{n}+\mathrm{1}} =\mathrm{f}\left(\mathrm{a}_{\mathrm{n}} \right);\mathrm{f}\left(\mathrm{x}\right)=\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)\:\mathrm{increase}\:\mathrm{function} \\…
Question Number 204264 by SANOGO last updated on 10/Feb/24 $$\underset{} {\int}{ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right){dx} \\ $$ Answered by witcher3 last updated on 10/Feb/24 $$\mathrm{by}\:\mathrm{part}\:\mathrm{u}=\mathrm{ln}\left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right);\mathrm{v}'\left(\mathrm{x}\right)=\mathrm{1} \\ $$$$=\mathrm{xln}\left(\mathrm{1}−\mathrm{x}^{\mathrm{2}}…
Question Number 204249 by universe last updated on 10/Feb/24 Commented by Frix last updated on 10/Feb/24 $${P}\left({x}\right)=\frac{\mathrm{5}{x}^{\mathrm{7}} }{\mathrm{16}}−\frac{\mathrm{21}{x}^{\mathrm{5}} }{\mathrm{16}}+\frac{\mathrm{35}{x}^{\mathrm{3}} }{\mathrm{16}}−\frac{\mathrm{35}{x}}{\mathrm{16}} \\ $$$${Q}\left({x}\right)=\frac{\mathrm{5}{x}^{\mathrm{3}} }{\mathrm{16}}+\frac{\mathrm{5}{x}^{\mathrm{2}} }{\mathrm{4}}+\frac{\mathrm{29}{x}}{\mathrm{16}}+\mathrm{1} \\…
Question Number 204250 by Noorzai last updated on 10/Feb/24 Answered by AST last updated on 10/Feb/24 $${tan}\left(\mathrm{9}\right)+{tan}\left(\mathrm{81}\right)−\left({tan}\mathrm{27}+{tan}\mathrm{63}\right) \\ $$$$=\frac{{sin}\mathrm{9}}{{cos}\mathrm{9}}+\frac{{sin}\mathrm{81}}{{cos}\mathrm{81}}−\left(\frac{{sin}\mathrm{27}}{{cos}\mathrm{27}}+\frac{{sin}\mathrm{63}}{{cos}\mathrm{63}}\right) \\ $$$$=\frac{\mathrm{2}{sin}\left(\mathrm{9}+\mathrm{81}\right)=\mathrm{2}}{\mathrm{2}{cos}\mathrm{9}{cos}\mathrm{81}}−\frac{\mathrm{2}{sin}\left(\mathrm{27}+\mathrm{63}\right)=\mathrm{2}}{\mathrm{2}{cos}\left(\mathrm{27}\right){cos}\left(\mathrm{63}\right)} \\ $$$$=\frac{\mathrm{2}}{\mathrm{2}{sin}\left(\mathrm{9}\right){cos}\left(\mathrm{9}\right)}−\frac{\mathrm{2}}{\mathrm{2}{sin}\left(\mathrm{27}\right){cos}\left(\mathrm{27}\right)}=\frac{\mathrm{2}}{{sin}\left(\mathrm{18}\right)}−\frac{\mathrm{2}}{{sin}\left(\mathrm{54}\right)} \\ $$$${Let}\:\theta=\mathrm{18}°\Rightarrow{sin}\left(\mathrm{5}\theta\right)=\mathrm{16}{sin}^{\mathrm{5}}…
Question Number 204262 by DEGWE last updated on 10/Feb/24 Answered by Frix last updated on 10/Feb/24 $$\mathrm{Charles}−\mathrm{Ange}\:\mathrm{LAISANT}\:\left(\mathrm{1905}\right): \\ $$$$\int{f}^{−\mathrm{1}} \left({x}\right){dx}={xf}^{−\mathrm{1}} \left({x}\right)−\left({F}\circ{f}^{−\mathrm{1}} \right)\left({x}\right)+{C} \\ $$$$\mathrm{with}\:{F}\left({x}\right)=\int{f}\left({x}\right){dx} \\…
Question Number 204273 by mustafazaheen last updated on 10/Feb/24 $$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\left(\mathrm{x}−\mathrm{1}\right)^{\mathrm{ln}\left(\frac{\mathrm{2}}{\mathrm{4}}\right)} } \\ $$$$\mathrm{Domain}\:\mathrm{f}\left(\mathrm{x}\right)\:=? \\ $$ Answered by Mathspace last updated on 11/Feb/24 $${f}\left({x}\right)=\frac{\mathrm{1}}{\left({x}−\mathrm{1}\right)^{{ln}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)} }=\frac{\mathrm{1}}{\left({x}−\mathrm{1}\right)^{−{ln}\mathrm{2}} }…
Question Number 204236 by Panav last updated on 09/Feb/24 $$\boldsymbol{{Helpful}}\:\boldsymbol{{questionfor}}\:\boldsymbol{{Olympiads}},\:\boldsymbol{{Find}}\:\boldsymbol{{Sol}}^{\boldsymbol{{n}}} . \\ $$ Commented by Panav last updated on 09/Feb/24 Answered by JDamian last updated…
Question Number 204233 by Perelman last updated on 09/Feb/24 Answered by Frix last updated on 09/Feb/24 $$\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}}…