Question Number 212314 by Spillover last updated on 09/Oct/24
Answered by mehdee7396 last updated on 09/Oct/24
$${f}\left({x}\right)={ln}\left(\frac{\mathrm{2}−{sinx}}{\mathrm{2}+{sinx}}\right) \\ $$$${f}\left(−{x}\right)={ln}\left(\frac{\mathrm{2}+{sinx}}{\mathrm{2}−{sinx}}\right)=−{f}\left({x}\right) \\ $$$$\Rightarrow;{f};\:{is}\:\:;{odd}\Rightarrow\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} {f}\left({x}\right){dx}=\mathrm{0}\:\checkmark \\ $$$$ \\ $$
Answered by Ghisom last updated on 09/Oct/24
$${f}\left({x}\right)=\mathrm{ln}\:\frac{\mathrm{2}−\mathrm{sin}\:{x}}{\mathrm{2}+\mathrm{sin}\:{x}} \\ $$$${f}\left(−{x}\right)=\mathrm{ln}\:\frac{\mathrm{2}+\mathrm{sin}\:{x}}{\mathrm{2}−\mathrm{sin}\:{x}}\:=−\mathrm{ln}\:\frac{\mathrm{2}−\mathrm{sin}\:{x}}{\mathrm{2}+\mathrm{sin}\:{x}}\:=−{f}\left({x}\right) \\ $$$$\Rightarrow\:\underset{−{a}} {\overset{{a}} {\int}}{f}\left({x}\right){dx}=\mathrm{0} \\ $$