Question Number 74019 by mathmax by abdo last updated on 17/Nov/19 $${let}\:{the}\:{matrix}\:\:{A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{2}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:−\mathrm{3}}\end{pmatrix} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}^{{n}} \:\:{for}\:{n}\:{integr} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{e}^{{A}} \:\:{and}\:{e}^{−{A}} . \\ $$ Commented by mathmax by…
Question Number 139555 by snipers237 last updated on 28/Apr/21 $$\:\:\underset{{i}=\mathrm{0}} {\overset{\mathrm{5}} {\prod}}\left(\mathrm{1}−{cotan}\left(\mathrm{20}+{i}\right)\right)\:\:\overset{?} {=}\:\mathrm{8} \\ $$ Answered by mr W last updated on 28/Apr/21 $$\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{tan}\:\mathrm{20}°}\right)\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{tan}\:\mathrm{25}°}\right) \\…
Question Number 74016 by mathmax by abdo last updated on 17/Nov/19 $${let}\:{g}\left({x}\right)\:=\frac{\mathrm{1}}{{x}}\int_{{x}} ^{\mathrm{2}{x}+\mathrm{1}} \:\:{arctan}\left({xt}\right){dt} \\ $$$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:{g}\left({x}\right)\:\:{and}\:{lim}_{{x}\rightarrow+\infty} {g}\left({x}\right). \\ $$ Commented by mathmax by abdo…
Question Number 74017 by mathmax by abdo last updated on 17/Nov/19 $${let}\:{f}\left({x}\right)=\int_{{x}} ^{{x}^{\mathrm{2}} +\mathrm{3}} \:{e}^{−{xt}} \:{ln}\left(\mathrm{1}+{e}^{−{xt}} \right){dt}\:\:\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:\:{lim}_{{x}\rightarrow+\infty} {f}\left({x}\right). \\ $$ Commented…
Question Number 74014 by mathmax by abdo last updated on 17/Nov/19 $$\:\: \\ $$$$\:\:{let}\:{W}\left({x}\right)=\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\frac{{x}^{{i}+{j}} }{{ij}} \\ $$$${calculate}\:{W}\:^{'} \left({x}\right). \\ $$ Commented by mathmax by…
Question Number 139548 by peter frank last updated on 28/Apr/21 Answered by Dwaipayan Shikari last updated on 28/Apr/21 $$\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \frac{{x}^{\mathrm{2}} {sin}\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{6}} }{dx}=−\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \frac{{x}^{\mathrm{2}}…
Question Number 74015 by mathmax by abdo last updated on 17/Nov/19 $${let}\:{f}\left({x}\right)\:=\int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\:\frac{{sh}\left({xt}\right)}{{sin}\left({xt}\right)}{dt} \\ $$$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} {f}\left({x}\right) \\ $$ Commented by mathmax by abdo…
Question Number 74013 by mathmax by abdo last updated on 17/Nov/19 $${let}\:\:\:{P}\left({x}\right)=\:\sum_{\mathrm{0}\leqslant{i}<{j}\leqslant{n}} \:{x}^{{i}+{j}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{P}\:^{'} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{P}\left({x}\right){dx} \\ $$ Commented by abdomathmax…
Question Number 139545 by aliibrahim1 last updated on 28/Apr/21 Answered by mr W last updated on 28/Apr/21 Commented by mr W last updated on 28/Apr/21…
Question Number 139544 by melanie last updated on 28/Apr/21 Answered by bemath last updated on 28/Apr/21 $$\:\mathrm{Tanzalin}\:\mathrm{formula} \\ $$$$\:\begin{array}{|c|c|c|c|}{\mathrm{u}\left(\mathrm{diff}\right)}&\hline{\mathrm{dv}\:\left(\mathrm{integrate}\right)}\\{\mathrm{4x}}&\hline{\mathrm{cos}\:\left(\mathrm{2}−\mathrm{3x}\right)}\\{\mathrm{4}}&\hline{−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{sin}\:\left(\mathrm{2}−\mathrm{3x}\right)}\\{\mathrm{0}}&\hline{−\frac{\mathrm{1}}{\mathrm{9}}\mathrm{cos}\:\left(\mathrm{2}−\mathrm{3x}\right)}\\\hline\end{array} \\ $$$$\mathrm{I}=\:−\frac{\mathrm{4x}\:\mathrm{sin}\:\left(\mathrm{2}−\mathrm{3x}\right)}{\mathrm{3}}\:+\frac{\mathrm{4}\:\mathrm{cos}\:\left(\mathrm{2}−\mathrm{3x}\right)}{\mathrm{9}}\:+\:\mathrm{c}\: \\ $$ Answered by…