Question Number 130748 by Dwaipayan Shikari last updated on 28/Jan/21 $$\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{3}} }+\frac{\mathrm{1}}{\mathrm{11}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{13}^{\mathrm{3}} }+\frac{\mathrm{1}}{\mathrm{19}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{21}^{\mathrm{3}} }+\frac{\mathrm{1}}{\mathrm{29}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{31}^{\mathrm{3}} }+.. \\ $$ Answered by mindispower last…
Question Number 65168 by Rio Michael last updated on 25/Jul/19 $${z}\:=\:\mathrm{1}−\:\mathrm{i}\sqrt{\mathrm{3}} \\ $$$${express}\:{z}\:{in}\:{the}\:{form}\:\:{r}\left({cos}\theta\:+\mathrm{i}{sin}\theta\right)\:{also}\:{express}\:{z}^{\mathrm{7}} \:{in}\:{the}\:{form} \\ $$$${re}^{\mathrm{i}\theta} . \\ $$ Answered by mr W last updated…
Question Number 65170 by Rio Michael last updated on 25/Jul/19 $${three}\:{forces}\:{F}_{\mathrm{1}} ,\:{F}_{\mathrm{2}} \:{and}\:{F}_{\mathrm{3}} \:{acts}\:{through}\:{the}\:{points}\:{with}\:{position}\:{vectors} \\ $$$$\boldsymbol{{r}}_{\mathrm{1}} ,{r}_{\mathrm{2}} \:{and}\:{r}_{\mathrm{3}} \:{respectively}\:{where} \\ $$$$\:{F}_{\mathrm{1}} \:=\left(\mathrm{3}{i}\:−\mathrm{2}{j}−\mathrm{4}{k}\right){N},\:\:\:{r}_{\mathrm{1}} =\:\left({i}\:+{k}\right){m} \\ $$$${F}_{\mathrm{2}}…
Question Number 65166 by Rio Michael last updated on 25/Jul/19 $${Given}\:{that}\:\:{f}\left({x}\right)\:=\:\frac{\mathrm{2}}{{x}^{\mathrm{2}} −\mathrm{1}} \\ $$$$\left.{a}\right)\:{Express}\:{f}\left({x}\right)\:{in}\:{partial}\:{fraction}. \\ $$$${b}.{Evaluate}\:\:\int_{\mathrm{3}} ^{\mathrm{5}} {f}\:\left({x}\right)\:{dx} \\ $$ Commented by mathmax by abdo…
Question Number 65113 by rajesh4661kumar@gamil.com last updated on 25/Jul/19 Commented by Prithwish sen last updated on 25/Jul/19 $$\mathrm{No}.\:\mathrm{of}\:\mathrm{term}\:\mathrm{on}\:\mathrm{n}^{\mathrm{tn}} \mathrm{group}\:=\:\mathrm{2n}−\mathrm{1} \\ $$$$\mathrm{The}\:\mathrm{1}^{\mathrm{st}\:} \mathrm{term}\:\mathrm{of}\:\mathrm{the}\:\mathrm{n}^{\mathrm{th}\:} \mathrm{group} \\ $$$$=\:\frac{\left(\mathrm{n}−\mathrm{1}\right)}{\mathrm{2}}\left[\mathrm{2}+\left(\mathrm{n}−\mathrm{2}\right)\mathrm{2}\right]\:+\mathrm{1}\:=\:\left(\mathrm{n}−\mathrm{1}\right)^{\mathrm{2}}…
Question Number 130604 by Study last updated on 27/Jan/21 $$\frac{{d}}{{dx}}\left({x}!\right)=??? \\ $$ Answered by Dwaipayan Shikari last updated on 27/Jan/21 $$\frac{{d}}{{dx}}\left({x}!\right)=\Gamma\left({x}+\mathrm{1}\right)\psi\left({x}+\mathrm{1}\right) \\ $$ Answered by…
Question Number 65052 by AnjanDey last updated on 24/Jul/19 $${A}.\mathrm{Evaluate}: \\ $$$$\left(\mathrm{i}\right)\int\frac{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}{\mathrm{9}+\mathrm{16sin}\:\mathrm{2}{x}}{dx} \\ $$$$\left(\mathrm{ii}\right)\int\frac{\mathrm{1}+{x}^{\mathrm{2}} }{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\sqrt{\mathrm{1}+{x}^{\mathrm{2}} +{x}^{\mathrm{4}} }}{dx} \\ $$$$\left(\mathrm{iii}\right)\int\frac{{x}−\mathrm{1}}{\left({x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{3}} +{x}+{x}^{\mathrm{2}} }}{dx} \\ $$ Answered…
Question Number 130555 by Dwaipayan Shikari last updated on 26/Jan/21 $$\frac{\mathrm{1}}{\left(\mathrm{2}−\pi\right)^{\mathrm{2}} }+\frac{\mathrm{1}}{\left(\mathrm{2}+\pi\right)^{\mathrm{2}} }+\frac{\mathrm{1}}{\left(\mathrm{6}−\pi\right)^{\mathrm{2}} }+\frac{\mathrm{1}}{\left(\mathrm{6}+\pi\right)^{\mathrm{2}} }+\frac{\mathrm{1}}{\left(\mathrm{10}−\pi\right)^{\mathrm{2}} }+\frac{\mathrm{1}}{\left(\mathrm{10}+\pi\right)^{\mathrm{2}} }+..=\frac{\pi^{\mathrm{2}} }{\mathrm{16}}{sec}^{\mathrm{2}} \left(\frac{\pi^{\mathrm{2}} }{\mathrm{4}}\right) \\ $$$${Prove}\:{or}\:{disprove} \\ $$ Answered…
Question Number 65013 by Rio Michael last updated on 24/Jul/19 $${why}\:{do}\:{we}\:{divide}\:{each}\:{term}\:{by}\:{n}\:{when}\:{given}\:{the}\:{question} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\:\frac{\mathrm{3}\:+\mathrm{2}{n}}{\mathrm{1}+{n}}\:? \\ $$ Answered by Tanmay chaudhury last updated on 24/Jul/19 $$\underset{{n}\rightarrow\infty}…
Question Number 65011 by AnjanDey last updated on 24/Jul/19 $$\mathrm{1}.\left(\mathrm{i}\right)\mathrm{Evaluate}:\int\frac{\mathrm{1}}{\mathrm{sin}\:{x}−\mathrm{cos}\:{x}+\sqrt{\mathrm{2}}}{dx} \\ $$$$\left(\mathrm{ii}\right)\mathrm{Evaluate}:\int\mathrm{2}^{\mathrm{2}^{\mathrm{2}^{{x}} } } \mathrm{2}^{\mathrm{2}^{{x}} } \mathrm{2}^{{x}} \:{dx} \\ $$$$\left(\mathrm{iii}\right)\mathrm{Evaluate}:\int\frac{\mathrm{cos}\:^{\mathrm{3}} {x}}{\mathrm{sin}\:^{\mathrm{2}} {x}+\mathrm{sin}\:{x}}{dx} \\ $$$$\mathrm{2}.\mathrm{cosec}\:\left[\mathrm{tan}^{−\mathrm{1}} \left\{\mathrm{cos}\:\left(\mathrm{cot}^{−\mathrm{1}}…