Question Number 163368 by nurtani last updated on 06/Jan/22
Answered by Ar Brandon last updated on 06/Jan/22
$${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\frac{\mathrm{4}}{\pi}\mathrm{sin}^{\mathrm{2007}} {x}}{\mathrm{sin}^{\mathrm{2007}} {x}+\mathrm{cos}^{\mathrm{2007}} {x}}{dx}…\left(\mathrm{1}\right) \\ $$$${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\frac{\mathrm{4}}{\pi}\mathrm{cos}^{\mathrm{2007}} {x}}{\mathrm{sin}^{\mathrm{2007}} {x}+\mathrm{cos}^{\mathrm{2007}} {x}}{dx}…\left(\mathrm{2}\right) \\ $$$$\left(\mathrm{1}\right)+\left(\mathrm{2}\right)\Rightarrow \\ $$$$\mathrm{2}{I}=\frac{\mathrm{4}}{\pi}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{sin}^{\mathrm{2007}} {x}+\mathrm{cos}^{\mathrm{2007}} {x}}{\mathrm{sin}^{\mathrm{2007}} {x}+\mathrm{cos}^{\mathrm{2007}} {x}}{dx}=\frac{\mathrm{4}}{\pi}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {dx} \\ $$$$\:\:{I}=\frac{\mathrm{2}}{\pi}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {dx}=\frac{\mathrm{2}}{\pi}\centerdot\frac{\pi}{\mathrm{2}}=\mathrm{1} \\ $$
Commented by nurtani last updated on 06/Jan/22
thanks mr brandon
Commented by amin96 last updated on 06/Jan/22
$$\left.{g}\left.{reat}\:{sir}\:{Brandon}.\:{Integral}\:{geniusss}\right)\right) \\ $$
Commented by Ar Brandon last updated on 06/Jan/22
My pleasure, friends!