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Category: Geometry

if-f-R-R-is-continuous-and-f-x-y-f-x-y-1-find-f-x-2-proof-or-disproof-that-f-x-1-3-if-f-0-0-proof-or-disproof-that-f-x-x-

Question Number 652 by 123456 last updated on 19/Feb/15 $${if}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:{is}\:{continuous}\:{and} \\ $$$${f}\left({x}+{y}\right)={f}\left({x}\right)+{y} \\ $$$$\mathrm{1}.\:{find}\:{f}\left({x}\right) \\ $$$$\mathrm{2}.\:{proof}\:{or}\:{disproof}\:{that}\:{f}'\left({x}\right)=\mathrm{1} \\ $$$$\mathrm{3}.\:{if}\:{f}\left(\mathrm{0}\right)=\mathrm{0}\:{proof}\:{or}\:{disproof}\:{that}\:{f}\left({x}\right)={x} \\ $$ Answered by prakash jain last…

pi-2-pi-2-sin-x-cos-x-dx-pi-2-pi-2-sin-x-cos-x-cos-2nx-dx-pi-2-pi-2-sin-x-cos-x-sin-2nx-dx-n-N-

Question Number 630 by 123456 last updated on 15/Feb/15 $$\underset{−\frac{\pi}{\mathrm{2}}} {\overset{+\frac{\pi}{\mathrm{2}}} {\int}}\frac{\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}}{dx} \\ $$$$\underset{−\frac{\pi}{\mathrm{2}}} {\overset{+\frac{\pi}{\mathrm{2}}} {\int}}\frac{\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}}\mathrm{cos}\left(\mathrm{2}{nx}\right){dx} \\ $$$$\underset{−\frac{\pi}{\mathrm{2}}} {\overset{+\frac{\pi}{\mathrm{2}}} {\int}}\frac{\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}}\mathrm{sin}\left(\mathrm{2}{nx}\right){dx} \\ $$$${n}\in\mathbb{N}^{\ast} \\ $$ Commented…

Question-66109

Question Number 66109 by Rio Michael last updated on 09/Aug/19 Commented by Rio Michael last updated on 09/Aug/19 $${The}\:{diagram}\:{above}\:{shows}\:{a}\:{uniform}\:{semi}−{circular}\:{lamina}\:{of}\:{radius}\:\mathrm{2}{a} \\ $$$$,{center}\:{O}.\:{The}\:{distance}\:{of}\:{the}\:{centre}\:{of}\:{mass}\:{form}\:{P},\:{vertically}\:{above}\:{O}\:{is} \\ $$$$ \\ $$$${A}\:\:\frac{\mathrm{6}{a}\pi−\mathrm{8}{a}}{\mathrm{3}\pi}…

proof-or-given-a-counter-example-if-p-q-are-prines-with-p-gt-q-and-s-prime-such-s-q-p-then-p-q-r-q-p-r-is-prime-r-

Question Number 510 by 123456 last updated on 21/Jan/15 $${proof}\:{or}\:{given}\:{a}\:{counter}\:{example}: \\ $$$${if}\:{p},{q}\:{are}\:{prines}\:{with}\:{p}>{q},\:{and}\:\exists{s}\:{prime} \\ $$$${such}\:{s}\in\left({q},{p}\right)\:{then} \\ $$$${p}−{q}\leqslant\underset{{r}\in\left({q},{p}\right),{r}\:{is}\:{prime}} {\sum}{r} \\ $$$$ \\ $$ Commented by prakash jain…

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}…