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Category: Integration

Question-37517

Question Number 37517 by ajfour last updated on 14/Jun/18 Commented by ajfour last updated on 14/Jun/18 $${Find}\:{volume}\:{of}\:{the}\:{region}\: \\ $$$${common}\:{to}\:{the}\:{two}\:{spheres}. \\ $$$$\left({The}\:{center}\:{of}\:{smaller}\:{sphere}\:{is}\right. \\ $$$${on}\:{the}\:{surface}\:{of}\:{the}\:{bigger} \\ $$$$\left.{sphere}\right).…

lim-x-0-x-2-sin-x-4-x-

Question Number 102987 by bobhans last updated on 12/Jul/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}^{\mathrm{2}} \:\mathrm{sin}\:\left({x}^{−\mathrm{4}} \right)}{{x}}\:?\: \\ $$ Answered by bemath last updated on 12/Jul/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{2}{x}\:\mathrm{sin}\:\left({x}^{−\mathrm{4}} \right)−\mathrm{4}{x}^{−\mathrm{3}}…

d-1-2-tan-2cot-2-d-2tan-2-1-2-cot-

Question Number 37432 by MJS last updated on 13/Jun/18 $$\int\frac{{d}\alpha}{\frac{\mathrm{1}}{\mathrm{2}}\mathrm{tan}\:\alpha\:+\mathrm{2cot}\:\frac{\alpha}{\mathrm{2}}}=? \\ $$$$\int\frac{{d}\beta}{\mathrm{2tan}\:\frac{\beta}{\mathrm{2}}\:+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cot}\:\beta}=? \\ $$ Answered by ajfour last updated on 13/Jun/18 $$\int\frac{\mathrm{2sin}\:\alpha\mathrm{cos}\:\alpha{d}\alpha}{\mathrm{sin}\:^{\mathrm{2}} \alpha+\mathrm{4cos}\:^{\mathrm{2}} \alpha}=\int\frac{\mathrm{sin}\:\mathrm{2}\alpha{d}\alpha}{\mathrm{1}+\frac{\mathrm{3}}{\mathrm{2}}\left(\mathrm{1}+\mathrm{cos}\:\mathrm{2}\alpha\right)} \\…

Question-168499

Question Number 168499 by infinityaction last updated on 12/Apr/22 Answered by MJS_new last updated on 12/Apr/22 $$\left(\mathrm{1}\right)\:{t}=\mathrm{tan}\:\frac{{x}}{\mathrm{2}} \\ $$$$\left(\mathrm{2}\right)\:{u}=\frac{\sqrt{\mathrm{1}−{t}}}{\:\sqrt{\mathrm{1}+{t}}} \\ $$$$\Rightarrow \\ $$$$−\int\left({u}^{\mathrm{10}} +{u}^{\mathrm{6}} \right){du}…