Question Number 497 by 13/NaSaNa(N)056565 last updated on 25/Jan/15 $$\int{e}^{{x}} \mathrm{sin}\:\mathrm{2}{x}\:{dx} \\ $$ Answered by prakash jain last updated on 16/Jan/15 $$\mathrm{Integrate}\:\mathrm{by}\:\mathrm{parts} \\ $$$${I}=\int{e}^{{x}} \mathrm{sin}\:\mathrm{2}{x}\:{dx}…
Question Number 487 by 13/NaSaNa(N)056565 last updated on 25/Jan/15 $$\int\frac{\mathrm{1}}{\mathrm{1}+\sqrt{\mathrm{2}{x}}}\:\:{dx} \\ $$ Answered by prakash jain last updated on 14/Jan/15 $${x}=\mathrm{2}{t}^{\mathrm{2}} \\ $$$${dx}=\mathrm{4}{t}\:{dt} \\ $$$$\int\frac{\mathrm{1}}{\mathrm{1}+\sqrt{\mathrm{2}{x}}}\:{dx}=\int\:\frac{\mathrm{4}{tdt}}{\mathrm{1}+\mathrm{2}{t}}…
Question Number 483 by 13/NaSaNa(N)056565 last updated on 25/Jan/15 $$\int\frac{\mathrm{1}}{\:\sqrt{{x}−\mathrm{1}}}{dx} \\ $$$$ \\ $$ Answered by 123456 last updated on 12/Jan/15 $$\int\frac{{dx}}{\:\sqrt{{x}−\mathrm{1}}} \\ $$$${u}={x}−\mathrm{1} \\…
Question Number 480 by 123456 last updated on 11/Jan/15 $${proof}\:{or}\:{given}\:{a}\:{counter}\:{example}: \\ $$$${for}\:{s}\in\left\{\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5}\right\} \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left[\frac{\mathrm{1}}{{s}^{{i}} }−\frac{\left(−\mathrm{1}\right)^{{i}} }{{i}^{{s}} }\right]\leqslant\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{s}+\mathrm{1}}{{si}^{{s}} } \\ $$ Commented…
Question Number 472 by 123456 last updated on 25/Jan/15 $${proof}\:{or}\:{give}\:{a}\:{counter}−{example}: \\ $$$${if}\:{nm}\:{is}\:{prime}\:{then}\:\mathrm{mdc}\left({n}^{\mathrm{2}} ,{m}^{\mathrm{2}} \right)=\mathrm{1} \\ $$ Answered by prakash jain last updated on 10/Jan/15 $$\mathrm{If}\:{n},{m}\:\in\mathbb{Z}\:\mathrm{then}\:{n}=\mathrm{1}\:\mathrm{or}\:{m}=\mathrm{1}…
Question Number 465 by shubham last updated on 09/Jan/15 $${A}\:{particle}\:{moves}\:{with}\:{a}\:{central}\:{acceration}\:{varies}\:{as}\:{the}\:{cube}\:{of}\:{the}\:{distance}.\:{if}\:{it}\:{be}\:{projected}\:{from}\:{an}\:{apse}\:{at}\:{distance}\:{a}\:{from}\:{the}\:{origin}\:{with}\:{a}\:{velocity}\:{which}\:{is}\:\sqrt{\mathrm{2}\:}\:{times}\:{the}\:{velocity}\:{for}\:{a}\:{circle}\:{of}\:{radius}\:{a}.\:{show}\:{that}\:{the}\:{equation}\:{of}\:{path}\:{is}\:{its}\:{r}\mathrm{cos}\theta/\sqrt{\mathrm{2}}\:=\:{a} \\ $$ Commented by prakash jain last updated on 09/Jan/15 $$\mathrm{Reformatted} \\ $$$${A}\:{particle}\:{moves}\:{with}\:{a}\:{central} \\ $$$${acceration}\:{varies}\:{as}\:{the}\:{cube}\:…
Question Number 131534 by mr W last updated on 06/Feb/21 Commented by mr W last updated on 05/Feb/21 $${find}\:{the}\:{radii}\:{of}\:{the}\:{inscribing}\:{and} \\ $$$${the}\:{circumscribing}\:{sphere}\:{of}\:{a} \\ $$$${tetrahedron}\:{with}\:{edge}\:{lengthes}\:{a},{b},{c}, \\ $$$${p},{q},{r}\:{as}\:{shown}.…
Question Number 65980 by Tony Lin last updated on 07/Aug/19 Commented by Tanmay chaudhury last updated on 07/Aug/19 Answered by Tanmay chaudhury last updated on…
Question Number 435 by novrya last updated on 03/Jan/15 $$\int\sqrt{{cos}\:{x}}\:{dx}\:=…. \\ $$$$ \\ $$ Commented by 123456 last updated on 03/Jan/15 $$\mathrm{2E}\left(\frac{{x}}{\mathrm{2}}\mid\mathrm{2}\right)+{C} \\ $$ Commented…
Question Number 433 by 123456 last updated on 25/Jan/15 $$\mathrm{find}\:\mathrm{tagent}\:\mathrm{plane}\:\mathrm{of}\:\mathrm{surface} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{3}} ={z}^{\mathrm{4}} \\ $$$$\mathrm{at}\:\mathrm{the}\:\mathrm{point}\:\left(\mathrm{28},\mathrm{8},\mathrm{6}\right) \\ $$ Answered by prakash jain last updated on…