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Question Number 32348 by abdo imad last updated on 23/Mar/18
![1)let n ∈Nand A_n = ∫_0 ^π (dx/(1+cos^2 (nx))) .calculate A_n 2) f∈ C^0 ([0,π], R) find lim_(n→∞) ∫_0 ^π ((f(x))/(1+cos^2 (nx)))dx .](Q32348.png)
$$\left.\mathrm{1}\right){let}\:{n}\:\in{Nand}\:\:\:{A}_{{n}} \:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dx}}{\mathrm{1}+{cos}^{\mathrm{2}} \left({nx}\right)}\:.{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{f}\in\:{C}^{\mathrm{0}} \left(\left[\mathrm{0},\pi\right],\:{R}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{f}\left({x}\right)}{\mathrm{1}+{cos}^{\mathrm{2}} \left({nx}\right)}{dx}\:. \\ $$