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Author: Tinku Tara

Question-221944

Question Number 221944 by hardmath last updated on 13/Jun/25 Commented by hardmath last updated on 13/Jun/25 $$\mathrm{If}: \\ $$$$\bigtriangleup\mathrm{ABC}\:-\:\mathrm{Equilateral}\:\mathrm{Triangle} \\ $$$$\mathrm{MK}\:\parallel\:\mathrm{AB} \\ $$$$\mathrm{MN}\:\parallel\:\mathrm{AC} \\ $$$$\mathrm{ML}\:\bot\:\mathrm{AB}…

Question-221943

Question Number 221943 by hardmath last updated on 13/Jun/25 Answered by MrGaster last updated on 13/Jun/25 $$\underset{{i}=\mathrm{1}} {\overset{{k}} {\sum}}{i}\left({k}−{i}+\frac{\mathrm{2}}{\mathrm{3}}\right)=\underset{{i}=\mathrm{1}} {\overset{{k}} {\sum}}\left({ki}−{i}^{\mathrm{2}} +\frac{\mathrm{2}}{\mathrm{3}}{i}\right)={k}\underset{{i}=\mathrm{1}} {\overset{{k}} {\sum}}{i}−\underset{{i}=\mathrm{1}} {\overset{{k}}…

sin-1-cos-x-dx-

Question Number 221957 by fantastic last updated on 13/Jun/25 $$\int\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{cos}\:{x}\right){dx} \\ $$ Answered by Ghisom last updated on 13/Jun/25 $$\int\mathrm{arcsin}\:\mathrm{cos}\:{x}\:{dx}= \\ $$$$\:\:\:\:\:\left[\mathrm{by}\:\mathrm{parts}:\:\begin{cases}{{u}'=\mathrm{1}\:\rightarrow\:{u}={x}}\\{{v}=\mathrm{arcsin}\:\mathrm{cos}\:{x}\:\rightarrow\:{v}'=−\sigma}\end{cases}\right] \\ $$$$\:\:\:\:\:\mathrm{where}\:\sigma=\mathrm{sign}\:\left(\mathrm{sin}\:{x}\right)…