Question Number 66140 by AnjanDey last updated on 09/Aug/19
$$\mathrm{1}.\boldsymbol{{Show}}\:\boldsymbol{{that}}:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left(\mathrm{sin}\:\mathrm{2}{x}\right)\mathrm{sin}\:{x}\:{dx}=\sqrt{\mathrm{2}}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {f}\left(\mathrm{cos}\:\mathrm{2}{x}\right)\mathrm{cos}\:{x}\:{dx}. \\ $$$$\mathrm{2}.\boldsymbol{{If}}\:\boldsymbol{{f}}\left(\boldsymbol{{z}}\right)=\frac{\boldsymbol{{d}}}{\boldsymbol{{dz}}}\left\{\mathrm{5}^{\mid\boldsymbol{{f}}\left(\boldsymbol{{z}}\right)\mid} \right\}\:\:\boldsymbol{{then}}\:\boldsymbol{{what}}\:\boldsymbol{{is}}\:\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\:\boldsymbol{{f}}'\left(\boldsymbol{{e}}\right)? \\ $$