Menu Close

Author: Tinku Tara

Solve-for-x-and-y-x-y-y-x-17-

Question Number 208783 by Tawa11 last updated on 22/Jun/24 $$\mathrm{Solve}\:\mathrm{for}\:\:\mathrm{x}\:\:\mathrm{and}\:\:\mathrm{y}. \\ $$$$\:\:\:\:\mathrm{x}^{\mathrm{y}} \:\:−\:\:\mathrm{y}^{\mathrm{x}} \:\:=\:\:\mathrm{17} \\ $$ Commented by Tawa11 last updated on 22/Jun/24 $$\mathrm{x}\:=\:\mathrm{3}\:\mathrm{and}\:\mathrm{y}\:=\:\mathrm{4}. \\…

sum-1-1-4-7-1-4-7-10-1-7-10-13-1-10-13-16-1-13-16-19-1-94-97-100-

Question Number 208738 by Tawa11 last updated on 22/Jun/24 $$\mathrm{sum}: \\ $$$$\:\:\:\frac{\mathrm{1}}{\mathrm{1}.\mathrm{4}.\mathrm{7}}\:\:\:+\:\:\frac{\mathrm{1}}{\mathrm{4}.\mathrm{7}.\mathrm{10}}\:\:+\:\:\frac{\mathrm{1}}{\mathrm{7}.\mathrm{10}.\mathrm{13}}\:\:+\:\:\frac{\mathrm{1}}{\mathrm{10}.\mathrm{13}.\mathrm{16}}\:\:+\:\:\frac{\mathrm{1}}{\mathrm{13}.\mathrm{16}.\mathrm{19}}\:\:+\:\:…\:\:+\:\:\frac{\mathrm{1}}{\mathrm{94}.\mathrm{97}.\mathrm{100}} \\ $$ Answered by BaliramKumar last updated on 22/Jun/24 $$\frac{\mathrm{1}}{\mathrm{6}}\left[\frac{\mathrm{1}}{\mathrm{1}×\mathrm{4}}\:−\:\frac{\mathrm{1}}{\mathrm{97}×\mathrm{100}}\right]\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{d}=\:\mathrm{7}−\mathrm{1}\:=\:\mathrm{10}−\mathrm{4}\:=\:\mathrm{6} \\ $$$$\frac{\mathrm{1}}{\mathrm{6}}×\frac{\left(\mathrm{97}×\mathrm{25}−\mathrm{1}\right)}{\mathrm{97}×\mathrm{100}}\:=\:\frac{\mathrm{1}}{\mathrm{6}}×\frac{\mathrm{2424}}{\mathrm{9700}}\: \\…

0-pi-2-xln-sin-x-dx-

Question Number 208692 by Ghisom last updated on 21/Jun/24 $$\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}{x}\mathrm{ln}\:\mathrm{sin}\:{x}\:{dx}=? \\ $$ Answered by Berbere last updated on 21/Jun/24 $${ln}\left({sin}\left({x}\right)\right)=−{ln}\left(\mathrm{2}\right)−\underset{{k}\geqslant\mathrm{1}} {\sum}\frac{{cos}\left(\mathrm{2}{kx}\right)}{{k}} \\ $$$${proof}\:{ln}\left({sin}\left({x}\right)\right)={Re}\left({ln}\left({sin}\left({x}\right)\right)\right.…