Question Number 146361 by gsk2684 last updated on 13/Jul/21 $${find}\:\int\frac{\mathrm{1}}{{x}^{{n}} +\mathrm{1}}{dx}\:{for}\:{n}\in{N} \\ $$ Answered by mathmax by abdo last updated on 13/Jul/21 $$\mathrm{z}^{\mathrm{n}} \:+\mathrm{1}=\mathrm{0}\:\Rightarrow\mathrm{z}^{\mathrm{n}} \:=\mathrm{e}^{\left(\mathrm{2k}+\mathrm{1}\right)\mathrm{i}\pi}…
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Question Number 80770 by zainal tanjung last updated on 06/Feb/20 $$\int\mathrm{x}^{\mathrm{2}} +\mathrm{3x}\:\mathrm{dx}=.. \\ $$ Commented by zainal tanjung last updated on 06/Feb/20 $$\mathrm{you}\:\mathrm{are}\:\mathrm{right}.\:\mathrm{good}. \\ $$…
Question Number 15235 by arnabpapu550@gmail.com last updated on 08/Jun/17 $$\int\left(\mathrm{log}\:\sqrt{\mathrm{x}\:}\:\right)^{\mathrm{2}} \mathrm{dx}=? \\ $$ Answered by liday last updated on 18/Jun/17 $$\int\left({log}\sqrt{{x}}\right)^{\mathrm{2}} {dx}=\frac{\mathrm{1}}{\mathrm{4}}\int\left({logx}\right)^{\mathrm{2}} {dx}=\frac{\mathrm{1}}{\mathrm{4}}\left[{x}\left({logx}\right)^{\mathrm{2}} −\int{x}\centerdot\mathrm{2}{logx}\centerdot\frac{\mathrm{1}}{{x}}{dx}\right] \\…
Question Number 146307 by mnjuly1970 last updated on 12/Jul/21 $$ \\ $$$$\:\:\:\:\:\:\:\mathrm{S}_{{n}} \:=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\:\mathrm{1}}{\left.{k}\:\left({k}+\mathrm{2}\right){k}+\mathrm{4}\right)} \\ $$$$\:\:\:\:\:\:\:\:{lim}_{\:{n}\rightarrow\infty} \:\left(\:\:\mathrm{S}_{\:{n}} \:\right)\:=\:? \\ $$ Answered by mathmax by…
Question Number 80764 by M±th+et£s last updated on 06/Feb/20 $${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} {x}\:{arctanh}\left({e}^{−\alpha{x}} \right){dx}=\frac{\mathrm{7}\zeta\left(\mathrm{3}\right)}{\mathrm{8}\alpha^{\mathrm{2}} } \\ $$ Answered by ~blr237~ last updated on 06/Feb/20…
Question Number 80752 by bagjamath last updated on 06/Feb/20 Commented by john santu last updated on 06/Feb/20 $${x}−\mathrm{2010}=\:{t}\: \\ $$$${x}−\mathrm{2009}={t}+\mathrm{1} \\ $$$$\int_{−\mathrm{1}} ^{\mathrm{0}} \left({t}^{\mathrm{2}} +{t}\right){dt}\:=\:\frac{\mathrm{1}}{\mathrm{3}}{t}^{\mathrm{3}}…
Question Number 146290 by SLVR last updated on 12/Jul/21 Commented by SLVR last updated on 12/Jul/21 $${Dear}\:{Olaf}\:{sir}\:{how}\:{r}\:{is}\:{taken}.. \\ $$$${how}\:{f}\left({x}\right)\:{was}\:{constructed}??? \\ $$$${kindly}\:{explain}… \\ $$$$ \\ $$…
Question Number 146212 by puissant last updated on 12/Jul/21 $$\mathrm{K}=\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{3}} }}\mathrm{dx} \\ $$ Answered by Ar Brandon last updated on 12/Jul/21 $$\mathrm{I}=\int\frac{\mathrm{dx}}{\:\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{3}} }}=\int\frac{\mathrm{dx}}{\:\sqrt{\mathrm{1}+\left(\mathrm{x}^{\mathrm{3}/\mathrm{2}} \right)^{\mathrm{2}} }},\:\mathrm{u}=\mathrm{x}^{\mathrm{3}/\mathrm{2}}…
Question Number 146181 by mnjuly1970 last updated on 11/Jul/21 $$ \\ $$$$\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{2}^{\:{n}} \:\left({n}+\mathrm{1}\:\right)\:\left(\:{n}\:+\:\mathrm{2}\:\right)}\:=? \\ $$ Answered by Olaf_Thorendsen last updated on 11/Jul/21 $$\mathrm{S}\:=\:\underset{{n}=\mathrm{0}}…