Question Number 167740 by LEKOUMA last updated on 24/Mar/22
$${I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{\mathrm{1}}{\mathrm{cos}\:^{\mathrm{2}{n}+\mathrm{1}} {x}}{dx} \\ $$$${Prove}\:{by}\:{parts}\:{that}: \\ $$$$\mathrm{2}{nI}_{{n}} =\left(\mathrm{2}{n}−\mathrm{1}\right){I}_{{n}−\mathrm{1}} +\frac{\mathrm{2}^{{n}} }{\:\sqrt{\mathrm{2}}} \\ $$