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Given-f-x-3-16-0-1-f-x-dx-x-2-9-10-0-2-f-x-dx-x-2-0-3-f-x-dx-4-Solve-lim-t-0-2t-f-2-2-f-1-t-f-x-2-dx-1-cos-t-cosh-2t-cos-3t-




Question Number 31838 by Joel578 last updated on 15/Mar/18
Given  f(x) = (3/(16)) (∫_0 ^1  f(x)dx)x^2  − (9/(10))(∫_0 ^2  f(x)dx)x + 2(∫_0 ^3  f(x)dx) + 4  Solve  lim_(t→0)  ((2t + (∫_(f(2) + 2) ^(f^(−1) (t)) [f ′(x)]^2  dx))/(1 − cos t cosh 2t cos 3t))
$$\mathrm{Given} \\ $$$${f}\left({x}\right)\:=\:\frac{\mathrm{3}}{\mathrm{16}}\:\left(\int_{\mathrm{0}} ^{\mathrm{1}} \:{f}\left({x}\right){dx}\right){x}^{\mathrm{2}} \:−\:\frac{\mathrm{9}}{\mathrm{10}}\left(\int_{\mathrm{0}} ^{\mathrm{2}} \:{f}\left({x}\right){dx}\right){x}\:+\:\mathrm{2}\left(\int_{\mathrm{0}} ^{\mathrm{3}} \:{f}\left({x}\right){dx}\right)\:+\:\mathrm{4} \\ $$$$\mathrm{Solve} \\ $$$$\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}{t}\:+\:\left(\int_{{f}\left(\mathrm{2}\right)\:+\:\mathrm{2}} ^{{f}^{−\mathrm{1}} \left({t}\right)} \left[{f}\:'\left({x}\right)\right]^{\mathrm{2}} \:{dx}\right)}{\mathrm{1}\:−\:\mathrm{cos}\:{t}\:\mathrm{cosh}\:\mathrm{2}{t}\:\mathrm{cos}\:\mathrm{3}{t}} \\ $$

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