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Author: Tinku Tara

proof-or-given-a-counter-example-if-n-N-n-gt-1-exist-a-number-k-N-k-0-n-such-that-n-k-is-prime-

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-66036

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…

e-x-sin-2x-dx-

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}…

1-x-1-x-dx-

Question Number 492 by 123456 last updated on 25/Jan/15 $$\int\frac{\mathrm{1}−\sqrt{{x}}}{\mathrm{1}+\sqrt{{x}}}{dx} \\ $$ Answered by prakash jain last updated on 14/Jan/15 $${x}={t}^{\mathrm{2}} \\ $$$${dx}=\mathrm{2}{t}\:{dt} \\ $$$$\int\frac{\mathrm{1}−{t}}{\mathrm{1}+{t}}\centerdot\mathrm{2}{t}\:{dt}=\mathrm{2}\int\:\frac{{t}−{t}^{\mathrm{2}}…

1-1-2x-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}}…