Question Number 66211 by mr W last updated on 10/Aug/19 Commented by mr W last updated on 10/Aug/19 $${Find}\:{the}\:{maximum}\:{area}\:{of}\:{a}\:{right} \\ $$$${triangle}\:{inscribed}\:{in}\:{an}\:{ellipse}. \\ $$ Commented by…
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…
Question Number 650 by 123456 last updated on 19/Feb/15 $${f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${g}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${f}\left({x}+{y}\right)={f}\left({x}\right)+{f}\left({y}\right){g}\left({y}\right) \\ $$$${g}\left({x}+{y}\right)={f}\left({x}\right){g}\left({x}\right)+{g}\left({y}\right) \\ $$ Answered by prakash jain last updated on…
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 Number 66109 by Rio Michael last updated on 09/Aug/19 Commented by Rio Michael last updated on 09/Aug/19 $${The}\:{diagram}\:{above}\:{shows}\:{a}\:{uniform}\:{semi}−{circular}\:{lamina}\:{of}\:{radius}\:\mathrm{2}{a} \\ $$$$,{center}\:{O}.\:{The}\:{distance}\:{of}\:{the}\:{centre}\:{of}\:{mass}\:{form}\:{P},\:{vertically}\:{above}\:{O}\:{is} \\ $$$$ \\ $$$${A}\:\:\frac{\mathrm{6}{a}\pi−\mathrm{8}{a}}{\mathrm{3}\pi}…
Question Number 131624 by mr W last updated on 06/Feb/21 Commented by mr W last updated on 06/Feb/21 $${an}\:{old}\:{unsolved}\:{question} \\ $$ Commented by MJS_new last…
Question Number 516 by 112358 last updated on 25/Jan/15 $${Find}\:{all}\:{triangles}\:{with}\:{consecutive} \\ $$$${integer}\:{sides}\:{and}\:{having}\:{an}\:{angle}\:{twice} \\ $$$${another}\:{angle}. \\ $$ Commented by prakash jain last updated on 22/Jan/15 $${a}={n}…
Question Number 510 by 123456 last updated on 21/Jan/15 $${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…
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 $$\mathrm{tan}\:{x}={t} \\ $$$$\mathrm{sec}^{\mathrm{2}} {x}\:{dx}={dt}…
Question Number 497 by 13/NaSaNa(N)056565 last updated on 25/Jan/15 $$\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}…