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Question-155421

Question Number 155421 by SANOGO last updated on 30/Sep/21 Answered by puissant last updated on 30/Sep/21 $${Q}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{sint}}{\mathrm{1}+{cos}^{\mathrm{2}} {t}}{dt}\:=\:−\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{−{sint}}{\mathrm{1}+{cos}^{\mathrm{2}} {t}}{dt} \\ $$$$=−\left[{arctan}\left({cost}\right)\right]_{\mathrm{0}}…

Question-155399

Question Number 155399 by 0731619 last updated on 30/Sep/21 Answered by TheHoneyCat last updated on 30/Sep/21 $$\mathrm{lim}_{\mathrm{n}\rightarrow+\infty\:} \underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}}=+\infty \\ $$$$\mathrm{hence}\:\frac{\mathrm{1}}{\Sigma\frac{\mathrm{1}}{{n}}}\:\underset{{n}\rightarrow\infty} {\rightarrow}\mathrm{0} \\ $$…

Solve-x-2-5-i-x-12-5i-0-

Question Number 155384 by aliyn last updated on 29/Sep/21 $$\boldsymbol{{Solve}}\::\:\boldsymbol{{x}}^{\mathrm{2}} \:−\:\left(\mathrm{5}−\boldsymbol{{i}}\right)\boldsymbol{{x}}\:+\:\left(\mathrm{12}−\mathrm{5}\boldsymbol{{i}}\right)\:=\:\mathrm{0} \\ $$ Commented by aliyn last updated on 29/Sep/21 $$???? \\ $$ Answered by…

Question-155382

Question Number 155382 by MathsFan last updated on 29/Sep/21 Answered by puissant last updated on 29/Sep/21 $${Q}=\int_{\mathrm{0}} ^{\mathrm{10}} \sqrt{{x}+\sqrt{{x}+\sqrt{{x}+\sqrt{{x}+…}}}}{dx} \\ $$$${u}=\sqrt{{x}+\sqrt{{x}+\sqrt{{x}+\sqrt{{x}+….}}}}\rightarrow{u}^{\mathrm{2}} ={x}+\sqrt{{x}+\sqrt{{x}+\sqrt{{x}+….}}} \\ $$$$\Rightarrow\:{u}^{\mathrm{2}} −{u}={x}\:\Rightarrow\:{u}−{u}+\frac{\mathrm{1}}{\mathrm{4}}={x}+\frac{\mathrm{1}}{\mathrm{4}}…

y-10-2x-2-1-5-find-dy-dx-

Question Number 155365 by MathsFan last updated on 29/Sep/21 $$\:{y}=\frac{\mathrm{10}}{\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{5}} } \\ $$$$\:{find}\:\:\frac{{dy}}{{dx}} \\ $$ Answered by gsk2684 last updated on 29/Sep/21 $$\mathrm{10}\left(\frac{−\mathrm{5}}{\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{6}}…

Question-155360

Question Number 155360 by mathlove last updated on 29/Sep/21 Answered by puissant last updated on 30/Sep/21 $$\psi\left(\frac{\mathrm{3}}{\mathrm{2}}\right)=\psi\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\right)=\psi\left(\frac{\mathrm{1}}{\mathrm{2}}\right)+\frac{\mathrm{1}}{\frac{\mathrm{1}}{\mathrm{2}}}=\psi\left(\frac{\mathrm{1}}{\mathrm{2}}\right)+\mathrm{2} \\ $$$${or}\:\psi\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=−\mathrm{2}{ln}\mathrm{2}−\gamma \\ $$$$\Rightarrow\:\psi\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\:=\:−\mathrm{2}{ln}\mathrm{2}−\gamma+\mathrm{2}\:=\:\mathrm{2}+{ln}\frac{\mathrm{1}}{\mathrm{4}}−\gamma \\ $$ Terms of…

show-that-lim-x-0-sinx-x-1-

Question Number 155353 by zakirullah last updated on 29/Sep/21 $$\boldsymbol{{show}}\:\boldsymbol{{that}}\:\:\boldsymbol{{lim}}_{\boldsymbol{{x}}\rightarrow\mathrm{0}} \frac{\boldsymbol{{sinx}}}{\boldsymbol{{x}}}\:=\:\mathrm{1} \\ $$ Answered by puissant last updated on 29/Sep/21 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{sinx}}{{x}}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{sinx}−{sin}\mathrm{0}}{{x}−\mathrm{0}}={f}'\left(\mathrm{0}\right) \\ $$$${f}\left({x}\right)={sinx}\rightarrow{f}'\left({x}\right)={cosx}…