Question Number 191623 by Shrinava last updated on 27/Apr/23
Answered by mehdee42 last updated on 27/Apr/23
![let : f(x)=ax+b f(f(x))+f(x)=−x ⇒(a^2 +a)x+ab+2b=−x⇒a=((−1+(√3)i)/2)=e^(((2π)/3)i) & b=0⇒f(x)=e^(((2π)/3)i) x f(f(f((1/(2+cosx)))))=(1/(2+cosx)) ⇒Ω=∫_0 ^π (dx/(2+cosx))=∫_0 ^π ((1+tan(x/2))/(tan^2 (x/2) +3))dx let : tan(x/2)=u⇒Ω=2∫_0 ^∞ (du/(u^2 +3))=[(2/( (√3)))tan^(−1) ((u/( (√3))))]_0 ^∞ =(π/( (√3)))](https://www.tinkutara.com/question/Q191630.png)
$${let}\::\:{f}\left({x}\right)={ax}+{b} \\ $$$${f}\left({f}\left({x}\right)\right)+{f}\left({x}\right)=−{x}\:\Rightarrow\left({a}^{\mathrm{2}} +{a}\right){x}+{ab}+\mathrm{2}{b}=−{x}\Rightarrow{a}=\frac{−\mathrm{1}+\sqrt{\mathrm{3}}{i}}{\mathrm{2}}={e}^{\frac{\mathrm{2}\pi}{\mathrm{3}}{i}} \:\:\&\:\:{b}=\mathrm{0}\Rightarrow{f}\left({x}\right)={e}^{\frac{\mathrm{2}\pi}{\mathrm{3}}{i}} {x} \\ $$$${f}\left({f}\left({f}\left(\frac{\mathrm{1}}{\mathrm{2}+{cosx}}\right)\right)\right)=\frac{\mathrm{1}}{\mathrm{2}+{cosx}} \\ $$$$\Rightarrow\Omega=\int_{\mathrm{0}} ^{\pi} \frac{{dx}}{\mathrm{2}+{cosx}}=\int_{\mathrm{0}} ^{\pi} \:\frac{\mathrm{1}+{tan}\frac{{x}}{\mathrm{2}}}{{tan}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}\:+\mathrm{3}}{dx} \\ $$$${let}\::\:{tan}\frac{{x}}{\mathrm{2}}={u}\Rightarrow\Omega=\mathrm{2}\int_{\mathrm{0}} ^{\infty} \:\frac{{du}}{{u}^{\mathrm{2}} +\mathrm{3}}=\left[\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}{tan}^{−\mathrm{1}} \left(\frac{{u}}{\:\sqrt{\mathrm{3}}}\right)\right]_{\mathrm{0}} ^{\infty} \\ $$$$=\frac{\pi}{\:\sqrt{\mathrm{3}}} \\ $$
Answered by witcher3 last updated on 28/Apr/23
![f(f(0))+f(0)=0;f(0)=a f(f(x))+f(x)+x=0 (fof(x)+f(x)+x)o.f(x)−{fof(x)+f(x)+x}=0of(x)−0=0 fofof(x)+fof(x)+f(x)−fof(x)−f(x)−x=0 ⇒fofof(x)=x Ω=∫_0 ^π (1/(2+cos(x)))=∫_0 ^π (dx/(2cos^2 ((x/2))+1)) =2∫_0 ^(π/2) (dy/(cos^2 (y)(3+tg^2 (y)))) =(2/( (√3))).[tan^(−1) (((tg(y))/( (√3))))]_0 ^(π/2) =(π/( (√3)))](https://www.tinkutara.com/question/Q191638.png)
$$\mathrm{f}\left(\mathrm{f}\left(\mathrm{0}\right)\right)+\mathrm{f}\left(\mathrm{0}\right)=\mathrm{0};\mathrm{f}\left(\mathrm{0}\right)=\mathrm{a} \\ $$$$\mathrm{f}\left(\mathrm{f}\left(\mathrm{x}\right)\right)+\mathrm{f}\left(\mathrm{x}\right)+\mathrm{x}=\mathrm{0} \\ $$$$\left(\mathrm{fof}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{x}\right)+\mathrm{x}\right)\mathrm{o}.\mathrm{f}\left(\mathrm{x}\right)−\left\{\mathrm{fof}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{x}\right)+\mathrm{x}\right\}=\mathrm{0of}\left(\mathrm{x}\right)−\mathrm{0}=\mathrm{0} \\ $$$$\mathrm{fofof}\left(\mathrm{x}\right)+\mathrm{fof}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{x}\right)−\mathrm{fof}\left(\mathrm{x}\right)−\mathrm{f}\left(\mathrm{x}\right)−\mathrm{x}=\mathrm{0} \\ $$$$\Rightarrow\mathrm{fofof}\left(\mathrm{x}\right)=\mathrm{x} \\ $$$$ \\ $$$$\Omega=\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{1}}{\mathrm{2}+\mathrm{cos}\left(\mathrm{x}\right)}=\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{dx}}{\mathrm{2cos}^{\mathrm{2}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{1}} \\ $$$$=\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{dy}}{\mathrm{cos}^{\mathrm{2}} \left(\mathrm{y}\right)\left(\mathrm{3}+\mathrm{tg}^{\mathrm{2}} \left(\mathrm{y}\right)\right)} \\ $$$$=\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}.\left[\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{tg}\left(\mathrm{y}\right)}{\:\sqrt{\mathrm{3}}}\right)\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} =\frac{\pi}{\:\sqrt{\mathrm{3}}} \\ $$$$ \\ $$$$ \\ $$