Question Number 5835 by sanusihammed last updated on 31/May/16 $${Evaluate}\:{the}\:{integral}. \\ $$$$ \\ $$$$\int\left[\left({x}−\frac{{x}^{\mathrm{3}} }{\mathrm{2}}+\frac{{x}^{\mathrm{5}} }{\mathrm{2}.\mathrm{4}}−\frac{{x}^{\mathrm{7}} }{\mathrm{2}.\mathrm{4}.\mathrm{6}}+…\right)\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}^{\mathrm{2}} }+\frac{{x}^{\mathrm{4}} }{\mathrm{2}^{\mathrm{2}} .\mathrm{4}^{\mathrm{2}} }−\frac{{x}^{\mathrm{6}} }{\mathrm{2}^{\mathrm{2}} .\mathrm{4}^{\mathrm{2}} .\mathrm{6}^{\mathrm{2}}…
Question Number 71371 by rajesh4661kumar@gmail.com last updated on 14/Oct/19 Commented by prakash jain last updated on 15/Oct/19 $${x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{2}{y}−\mathrm{23}=\mathrm{0} \\ $$$$\left({x}−\mathrm{1}\right)^{\mathrm{2}} +\left({y}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{25} \\…
Question Number 5834 by sanusihammed last updated on 31/May/16 $${Find}\:{all}\:{positive}\:{integers}\:{n}\:{for}\:{which}\:{there}\:{exist} \\ $$$${non}−{negative}\:{integer}\:.\:{a}_{\mathrm{1}\:} {a}_{\mathrm{2}} \:{a}_{\mathrm{3}} \:…….\:{a}_{{n}} \:.\:{Such}\:{that} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}^{{a}_{\mathrm{1}} } }\:+\:\frac{\mathrm{1}}{\mathrm{2}^{{a}_{\mathrm{2}} } }\:+\:\frac{\mathrm{1}}{\mathrm{2}^{{a}_{\mathrm{3}} } }\:+\:….\:+\:\frac{\mathrm{1}}{\mathrm{2}^{{a}_{{n}} }…
Question Number 136907 by leena12345 last updated on 27/Mar/21 $$\int\sqrt{\frac{\mathrm{2}+{x}}{\mathrm{2}−{x}}}{dx} \\ $$ Answered by Mathspace last updated on 27/Mar/21 $${let}\:{I}=\int\sqrt{\frac{\mathrm{2}+{x}}{\mathrm{2}−{x}}}{dx}\:\:{changement} \\ $$$$\sqrt{\frac{\mathrm{2}+{x}}{\mathrm{2}−{x}}}={t}\:{give}\:\frac{\mathrm{2}+{x}}{\mathrm{2}−{x}}={t}^{\mathrm{2}\:} \:\Rightarrow \\ $$$$\mathrm{2}+{x}=\mathrm{2}{t}^{\mathrm{2}}…
Question Number 136906 by leena12345 last updated on 27/Mar/21 $$\underset{\mathrm{9}} {\overset{\infty} {\int}}\frac{\mathrm{1}}{\left({x}−\mathrm{8}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} }{dx} \\ $$ Answered by Dwaipayan Shikari last updated on 27/Mar/21 $$=−\mathrm{2}\left[\left({x}−\mathrm{8}\right)^{−\frac{\mathrm{1}}{\mathrm{2}}} \right]_{\mathrm{9}}…
Question Number 136897 by mohammad17 last updated on 27/Mar/21 Answered by Olaf last updated on 27/Mar/21 $$ \\ $$$${f}\left({z}\right)\:=\:{e}^{{i}\frac{\pi}{\mathrm{4}}} {z}+\left(\mathrm{1}−\mathrm{2}{i}\right) \\ $$$${f}\left({z}\right)−{z}_{\mathrm{0}} \:=\:{e}^{{i}\frac{\pi}{\mathrm{4}}} \left({z}−{z}_{\mathrm{0}} \right)+\left({e}^{{i}\frac{\pi}{\mathrm{4}}}…
Question Number 71362 by malwaan last updated on 14/Oct/19 $$\boldsymbol{\mathrm{please}}\:\boldsymbol{{prove}}\:\boldsymbol{{that}} \\ $$$$\underset{\boldsymbol{{x}}\rightarrow\mathrm{0}} {\boldsymbol{{lim}}}\frac{\boldsymbol{{x}}−\boldsymbol{{sinx}}}{\boldsymbol{{x}}^{\mathrm{3}} }\:=\frac{\mathrm{1}}{\mathrm{6}}\:\boldsymbol{{by}}\:\boldsymbol{{using}} \\ $$$$\boldsymbol{{x}}=\mathrm{3}\boldsymbol{{y}}\:\boldsymbol{{and}}\: \\ $$$$\boldsymbol{{sin}}\mathrm{3}\boldsymbol{{y}}=\mathrm{3}\boldsymbol{{siny}}−\mathrm{4}\boldsymbol{{sin}}^{\mathrm{3}} \boldsymbol{{y}} \\ $$ Terms of Service Privacy…
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Question Number 136899 by BHOOPENDRA last updated on 27/Mar/21 Answered by Olaf last updated on 28/Mar/21 $$\mathrm{3}. \\ $$$${f}\left({x}\right)\:=\:{x}^{\mathrm{2}} ,\:−\mathrm{2}\leqslant{x}\leqslant\mathrm{2} \\ $$$${a}_{\mathrm{0}} \left({f}\right)\:=\:\frac{\mathrm{1}}{\mathrm{T}}\int_{−\frac{\mathrm{T}}{\mathrm{2}}} ^{+\frac{\mathrm{T}}{\mathrm{2}}} {f}\left({x}\right){dx}…
Question Number 71360 by 20190927 last updated on 14/Oct/19 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\sqrt{\mathrm{1}+\mathrm{2x}^{\mathrm{4}} }\mathrm{cos}\:\left(\sqrt{\mathrm{2}}\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{5}} \mathrm{ln}\:\left(\mathrm{1}−\mathrm{2x}^{\mathrm{3}} \right)} \\ $$ Commented by mathmax by abdo last updated on…