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Question-221791

Question Number 221791 by hardmath last updated on 10/Jun/25 Commented by hardmath last updated on 10/Jun/25 $$\mathrm{ABC}\:-\:\mathrm{equilateral}\:\mathrm{triangle} \\ $$$$\mathrm{MK}\:\parallel\:\mathrm{AB} \\ $$$$\mathrm{MN}\:\parallel\:\mathrm{AC} \\ $$$$\mathrm{ML}\:\bot\:\mathrm{AB} \\ $$$$\mathrm{AC}\:=\:\mathrm{a}…

1-4x-1-4x-2-2-1-8x-2-x-

Question Number 221787 by efronzo1 last updated on 10/Jun/25 $$\:\:\sqrt{\frac{\mathrm{1}−\mathrm{4x}\sqrt{\mathrm{1}−\mathrm{4x}^{\mathrm{2}} }}{\mathrm{2}}}\:=\:\mathrm{1}−\mathrm{8x}^{\mathrm{2}} \\ $$$$\:\:\mathrm{x}=?\: \\ $$ Answered by mr W last updated on 11/Jun/25 $$\mathrm{1}−\mathrm{4}{x}^{\mathrm{2}} =\mathrm{1}−\left(\mathrm{2}{x}\right)^{\mathrm{2}}…

Question-221814

Question Number 221814 by Tawa11 last updated on 10/Jun/25 Answered by wewji12 last updated on 11/Jun/25 $$\mathrm{log}_{{a}} \left({b}\right)=\frac{\mathrm{log}_{{b}} \left({c}\right)}{\mathrm{log}_{{a}} \left({c}\right)} \\ $$$$\frac{\mathrm{ln}\left(\mathrm{2}\right)}{\mathrm{ln}\left({x}+\mathrm{1}\right)}+\frac{\mathrm{ln}\left(\mathrm{3}\right)}{\mathrm{ln}\left({x}+\mathrm{2}\right)}+\frac{\mathrm{ln}\left(\mathrm{4}\right)}{\mathrm{ln}\left({x}+\mathrm{3}\right)}=\mathrm{3} \\ $$$$\mathrm{if}\:{x}=\mathrm{1}\: \\…

For-n-N-n-3-Prove-0-1-2-lt-p-2n-e-2piip-2-e-4piin-d-gt-0-p-is-a-prime-number-

Question Number 221778 by MrGaster last updated on 10/Jun/25 $$\mathrm{For}\:\forall{n}\in\boldsymbol{{N}}^{\ast} ,{n}\geq\mathrm{3}\:\mathrm{Prove}: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\underset{\mathrm{2}<{p}\leq\mathrm{2}{n}} {\sum}{e}^{\mathrm{2}\pi{ip}+\alpha} \right)^{\mathrm{2}} {e}^{−\mathrm{4}\pi{in}\alpha} {d}\alpha>\mathrm{0},{p}\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number} \\ $$ Terms of Service Privacy…

0-pi-a-a-cos-2n-x-dx-a-gt-1-0-pi-2-2-cos-4-x-dx-lim-m-0-pi-cos-2n-2mx-a-cos-2n-x-dx-a-gt-1-n-N-

Question Number 221769 by MrGaster last updated on 10/Jun/25 $$\int_{\mathrm{0}} ^{\pi} \frac{{a}}{{a}−\mathrm{cos}^{\mathrm{2}{n}} {x}}{dx}=?,{a}>\mathrm{1} \\ $$$$\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{2}}{\mathrm{2}−\mathrm{cos}^{\mathrm{4}} {x}}{dx}=? \\ $$$$\underset{{m}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{cos}^{\mathrm{2}{n}} \left(\mathrm{2}{mx}\right)}{{a}−\mathrm{cos}^{\mathrm{2}{n}} {x}}{dx}=?,{a}>\mathrm{1},{n}\in\mathbb{N}^{+}…