Question Number 506 by Karting7442 last updated on 25/Jan/15 $${write}\:{each}\:{mixed}\:{number}\:{as}\:{a}\:{decimal}.\: \\ $$$$\:\:\:\:\:\mathrm{4}\frac{\mathrm{1}}{\mathrm{5}} \\ $$$$ \\ $$$$ \\ $$ Answered by prakash jain last updated on…
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Question Number 131573 by mathlove last updated on 06/Feb/21 $$\:{if}\:\:\:\:{f}\left({x}+\frac{\mathrm{1}}{{x}}\right)={x}^{\mathrm{3}} +\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\:\:\:\:\:{then}\:\:{faind}\:\: \\ $$$${f}\left(\frac{\mathrm{1}}{{x}}\right)=? \\ $$ Answered by rs4089 last updated on 06/Feb/21 $${f}\left({x}+\frac{\mathrm{1}}{{x}}\right)={x}^{\mathrm{3}} +\frac{\mathrm{1}}{{x}^{\mathrm{3}}…
Question Number 503 by 123456 last updated on 20/Jan/15 $${proof}\:{or}\:{given}\:{a}\:{counter}−{example}: \\ $$$${if}\:{n}\in\mathbb{N},{n}>\mathrm{1},\:{exist}\:{a}\:{number}\:{k}\in\mathbb{N} \\ $$$${k}\in\left(\mathrm{0},{n}\right]\:{such}\:{that}\:{n}+{k}\:{is}\:{prime}. \\ $$ Commented by prakash jain last updated on 20/Jan/15 $$\mathrm{Bertrand}'\mathrm{s}\:\mathrm{theorem}\:\mathrm{states}\:\mathrm{that}\:…
Question Number 502 by 123456 last updated on 25/Jan/15 $$\underset{−\mathrm{1}} {\overset{+\mathrm{1}} {\int}}\frac{\mathrm{ln}\:\mid{x}\mid}{{x}^{\mathrm{2}} −\mathrm{1}}{dx} \\ $$ Answered by prakash jain last updated on 20/Jan/15 $${I}=\underset{−\mathrm{1}} {\overset{+\mathrm{1}}…
Question Number 66036 by aliesam last updated on 08/Aug/19 Answered by mr W last updated on 08/Aug/19 $$=\mathrm{2}\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+…+\frac{{n}}{\mathrm{2}}\right) \\ $$$$=\mathrm{2}×\frac{\left(\mathrm{1}+\frac{{n}}{\mathrm{2}}\right)\frac{{n}}{\mathrm{2}}}{\mathrm{2}} \\ $$$$=\frac{{n}\left({n}+\mathrm{2}\right)}{\mathrm{4}} \\ $$ Terms…
Question Number 499 by 123456 last updated on 18/Jan/15 $$\underset{−\infty} {\overset{\mathrm{0}} {\int}}\:\frac{{e}^{{x}} \mathrm{ln}\:\left(−{x}\right)}{{x}}{dx} \\ $$ Commented by prakash jain last updated on 19/Jan/15 $$\mathrm{Doesn}'\mathrm{t}\:\mathrm{converge}. \\…
Question Number 66032 by mathmax by abdo last updated on 07/Aug/19 $${simplify}\:\:{w}_{{n}} =\left(\mathrm{1}+{in}\right)^{{n}} −\left(\mathrm{1}−{in}\right)^{{n}} \:{with}\:{n}\:{integr}\:{natural} \\ $$ Commented by mr W last updated on 08/Aug/19…
Question Number 497 by 13/NaSaNa(N)056565 last updated on 25/Jan/15 $$\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} \\ $$$${I}=\int{e}^{{x}} \mathrm{sin}\:\mathrm{2}{x}\:{dx}…
Question Number 495 by 13/NaSaNa(N)056565 last updated on 25/Jan/15 $$\int\mathrm{sec}^{\mathrm{3}} {xdx} \\ $$ Answered by prakash jain last updated on 15/Jan/15 $$\mathrm{tan}\:{x}={t} \\ $$$$\mathrm{sec}^{\mathrm{2}} {x}\:{dx}={dt}…