Question Number 212050 by Spillover last updated on 28/Sep/24 Answered by Ghisom last updated on 28/Sep/24 $$\int\frac{{x}}{\:\sqrt{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{1}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\right)}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\mathrm{2}\sqrt{\mathrm{1}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}\right] \\ $$$$=\int{dt}={t}= \\…
Question Number 211979 by Spillover last updated on 25/Sep/24 Answered by BHOOPENDRA last updated on 25/Sep/24 $$\int\frac{{dx}}{\left({x}^{\mathrm{2}} \mathrm{tan}^{−\mathrm{1}} {x}+\mathrm{tan}^{−\mathrm{1}} {x}\:+{x}^{\mathrm{2}} \pi+\pi\right)} \\ $$$$\int\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{tan}^{−\mathrm{1}} {x}+\pi\right)}…
Question Number 211920 by Spillover last updated on 24/Sep/24 Answered by aleks041103 last updated on 24/Sep/24 $$\sqrt{{x}\sqrt{{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{3}} \sqrt{…}}}\:}=\:{x}^{\mathrm{1}/\mathrm{2}} {x}^{\mathrm{2}/\mathrm{4}} {x}^{\mathrm{3}/\mathrm{8}} …{x}^{{n}/\mathrm{2}^{{n}} } …\:= \\…
Question Number 211895 by Spillover last updated on 23/Sep/24 Answered by mehdee7396 last updated on 23/Sep/24 $${lim}_{{x}\rightarrow\infty} \:\left(\mathrm{1}+\frac{{a}}{{x}}−\frac{\mathrm{4}}{{x}^{\mathrm{2}} }−\mathrm{1}\right)\mathrm{2}{x} \\ $$$$={lim}_{{x}\rightarrow\infty} \:\left(\frac{{ax}−\mathrm{4}}{{x}^{\mathrm{2}} }\right)\mathrm{2}{x}=\mathrm{2}{a} \\ $$$$\Rightarrow{lim}_{{x}\rightarrow\infty}…
Question Number 211871 by Spillover last updated on 22/Sep/24 Answered by Ghisom last updated on 23/Sep/24 $$=−\underset{\mathrm{0}} {\overset{\mathrm{arccos}\:\frac{\sqrt{\mathrm{6}}}{\mathrm{3}}} {\int}}\frac{\mathrm{2}+\mathrm{tan}\:{x}}{\mathrm{3}−\mathrm{2cos}^{\mathrm{2}} \:{x}\:−\mathrm{sin}^{\mathrm{2}} \:{x}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt{\mathrm{2}}\mathrm{tan}\:{x}\right] \\ $$$$=−\frac{\mathrm{1}}{\mathrm{2}}\underset{\mathrm{0}}…
Question Number 211873 by Spillover last updated on 22/Sep/24 Answered by IbtisamAdnan last updated on 23/Sep/24 $$\:\:\:\boldsymbol{\mathrm{lim}}_{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{4}} \frac{\left(\boldsymbol{\mathrm{cos}\alpha}\right)^{\mathrm{x}} −\left(\mathrm{sin}\alpha\right)^{\mathrm{x}} −\mathrm{cos}\:\mathrm{2}\alpha}{\mathrm{x}−\mathrm{4}} \\ $$$$\boldsymbol{\mathrm{lim}}_{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{4}} \:\frac{\left(\boldsymbol{\mathrm{cos}\alpha}\right)^{\mathrm{x}} .\:\mathrm{ln}\:\mathrm{cos}\alpha\:\:−\:\left(\mathrm{sin}\:\alpha\right)^{\mathrm{x}} .\mathrm{ln}\:\mathrm{sin}\alpha}{\mathrm{1}}\left[\mathrm{L}\:\mathrm{hospital}\:\mathrm{rule}\right]…
Question Number 211734 by BaliramKumar last updated on 19/Sep/24 Answered by TonyCWX08 last updated on 19/Sep/24 $${y}={kx}\:\Rightarrow\:{k}=\frac{{y}}{{x}} \\ $$$${z}={ky}\:\Rightarrow\:{k}=\frac{{z}}{{y}} \\ $$$$\frac{{y}}{{x}}=\frac{{z}}{{y}} \\ $$$${y}^{\mathrm{2}} ={xy}\:=\:{B} \\…
Question Number 211636 by BaliramKumar last updated on 15/Sep/24 Answered by Rasheed.Sindhi last updated on 15/Sep/24 $$\blacktriangleright{A}={x}^{\mathrm{8}{k}+\mathrm{3}} +{x}^{\mathrm{8}{k}+\mathrm{6}} +{x}^{\mathrm{8}{k}+\mathrm{9}} +{x}^{\mathrm{8}{k}+\mathrm{12}} \\ $$$$\:\:\:\:\:\:\:\:={x}^{\mathrm{8}{k}+\mathrm{3}} \left(\mathrm{1}+{x}^{\mathrm{3}} +{x}^{\mathrm{6}} +{x}^{\mathrm{9}}…
Question Number 211457 by BaliramKumar last updated on 09/Sep/24 Answered by A5T last updated on 09/Sep/24 $$\sqrt[{\mathrm{7}}]{\mathrm{3}^{\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+\mathrm{5}+\mathrm{6}+\mathrm{7}} }=\mathrm{3}^{\mathrm{4}} =\mathrm{81} \\ $$ Answered by BHOOPENDRA last…
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