Question Number 139002 by BHOOPENDRA last updated on 21/Apr/21
Commented by Dwaipayan Shikari last updated on 22/Apr/21
$${Probability}\:\:\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \mid\Psi\left({x}\right)\mid^{\mathrm{2}} {dx}=\mathrm{1} \\ $$$$=\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} {A}^{\mathrm{2}} {cos}^{\mathrm{4}} {xdx}={A}^{\mathrm{2}} \Gamma\left(\frac{\mathrm{5}}{\mathrm{2}}\right)\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)={A}^{\mathrm{2}} \frac{\mathrm{3}\pi}{\mathrm{4}} \\ $$$$\Rightarrow\mathrm{1}={A}^{\mathrm{2}} \frac{\mathrm{3}\pi}{\mathrm{4}}\Rightarrow{A}=\pm\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}\pi}} \\ $$
Commented by Dwaipayan Shikari last updated on 22/Apr/21
$$\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} {A}^{\mathrm{2}} {cos}^{\mathrm{4}} {x}\:{dx}=\mathrm{1} \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {A}^{\mathrm{2}} {cos}^{\mathrm{4}} {x}\:{dx}=\frac{\mathrm{1}}{\mathrm{2}}\:\Rightarrow\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {A}^{\mathrm{2}} {cos}^{\mathrm{4}} {xdx}=\frac{\mathrm{1}}{\mathrm{4}}\:\:{or}\:\mathrm{25\%} \\ $$$${Probability}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$