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Author: Tinku Tara

nice-calculus-Evaluate-0-1-sin-ln-x-ln-x-ln-2-x-dx-Ans-ln-2-pi-4-1-

Question Number 126465 by mnjuly1970 last updated on 20/Dec/20 $$\:\:\:\:\:\:\:\:\:\:\:\:…\:{nice}\:\:{calculus}… \\ $$$$\:\:\:\:\:\mathscr{E}{valuate}\:… \\ $$$$\:\:\:\:\:\:\:\:\phi\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{sin}\left({ln}\left({x}\right)\right)−{ln}\left({x}\right)}{{ln}^{\mathrm{2}} \left({x}\right)}{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathscr{A}{ns}\:::\:{ln}\left(\sqrt{\mathrm{2}}\:\right)+\frac{\pi}{\mathrm{4}}\:−\mathrm{1}\:… \\ $$ Commented by talminator2856791 last…

Show-that-C-1-1-where-1-with-addition-operation-is-a-group-Help-

Question Number 191997 by Mastermind last updated on 05/May/23 $$\mathrm{Show}\:\mathrm{that}\:\mathbb{C}=\left\{−\mathrm{1},\mathrm{1},−\imath,\imath\right\}\:\mathrm{where} \\ $$$$\imath=\sqrt{−\mathrm{1}}\:\mathrm{with}\:\mathrm{addition}\:\mathrm{operation}\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{group}. \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$ Commented by Rasheed.Sindhi last updated…

Let-Fibonacci-sequence-F-n-n-0-where-F-0-0-F-1-1-and-F-n-2-F-n-1-F-n-n-0-Find-the-least-of-natural-numbers-n-so-that-F-n-and-F-n-1-1-

Question Number 60915 by naka3546 last updated on 27/May/19 $${Let}\:\:{Fibonacci}\:\:{sequence}\:\:\left({F}_{{n}} \right)\:_{{n}\geqslant\mathrm{0}} \\ $$$${where}\:\:{F}_{\mathrm{0}} \:=\:\mathrm{0},\:{F}_{\mathrm{1}} \:=\:\mathrm{1},\:\:{and}\:\:{F}_{{n}+\mathrm{2}} \:\:=\:\:{F}_{{n}+\mathrm{1}} \:+\:{F}_{{n}} \:\:\:\:,\:\:\forall\:{n}\:\:\geqslant\:\:\mathrm{0}\:. \\ $$$${Find}\:\:{the}\:\:{least}\:\:{of}\:\:{natural}\:\:{numbers}\:\:{n}\:\:{so}\:\:{that} \\ $$$${F}_{{n}} \:\:\:{and}\:\:\:{F}_{{n}+\mathrm{1}} \:−\:\mathrm{1}\:\:\:{can}\:\:{be}\:\:{divided}\:\:{by}\:\:\:{F}_{\mathrm{2019}} \:.…

Ques-1-Let-G-be-a-group-then-show-that-for-each-a-G-a-unique-element-e-G-a-e-e-a-a-Ques-2-If-a-G-x-G-and-x-is-unique-show-that-if-x-a-e-then-a-x-e-Hello-

Question Number 191986 by Mastermind last updated on 04/May/23 $$\mathrm{Ques}.\:\mathrm{1} \\ $$$$\mathrm{Let}\:\left(\mathrm{G},\ast\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{group},\:\mathrm{then}\:\mathrm{show} \\ $$$$\mathrm{that}\:\mathrm{for}\:\mathrm{each}\:\mathrm{a}\in\mathrm{G},\:\exists\:\mathrm{a}\:\mathrm{unique}\: \\ $$$$\mathrm{element}\:\mathrm{e}\in\mathrm{G}\:\mid\:\mathrm{a}\ast\mathrm{e}=\mathrm{e}\ast\mathrm{a}=\mathrm{a} \\ $$$$ \\ $$$$\mathrm{Ques}.\:\mathrm{2} \\ $$$$\mathrm{If}\:\mathrm{a}\in\mathrm{G}\:\Rightarrow\:\mathrm{x}\in\mathrm{G}\:\mathrm{and}\:\mathrm{x}\:\mathrm{is}\:\mathrm{unique} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{if}\:\mathrm{x}\ast\mathrm{a}=\mathrm{e},\:\mathrm{then}\:\mathrm{a}\ast\mathrm{x}=\mathrm{e}. \\…