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let-f-x-0-pi-2-ln-cos-xsin-d-with-x-fromR-1-determine-a-explicit-form-for-f-x-2-calculate-0-pi-2-ln-cos-sin-d-and-0-pi-2-ln-cos-2sin-d-




Question Number 59052 by maxmathsup by imad last updated on 04/May/19
let f(x) =∫_0 ^(π/2) ln(cosθ +xsinθ)dθ   with x fromR  1) determine a explicit form  for f(x)  2) calculate ∫_0 ^(π/2) ln(cosθ +sinθ) dθ  and ∫_0 ^(π/2) ln(cosθ +2sinθ)dθ .
$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({cos}\theta\:+{xsin}\theta\right){d}\theta\:\:\:{with}\:{x}\:{fromR} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({cos}\theta\:+{sin}\theta\right)\:{d}\theta\:\:{and}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({cos}\theta\:+\mathrm{2}{sin}\theta\right){d}\theta\:. \\ $$

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