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Advanced-Calculus-Evaluate-0-sec-4tan-2-5-d-L-rD-sE-GooD-LucK-




Question Number 118491 by Lordose last updated on 18/Oct/20
  ... ⧫Advanced Calculus⧫...    Evaluate::    Ω = ∫_0 ^( ∞) ((secθ)/( (√(4tan^2 θ+5))))dθ    ...♠L𝛗rD ∅sE♠...    ...♣GooD LucK♣
$$ \\ $$$$…\:\blacklozenge\mathrm{Advanced}\:\mathrm{Calculus}\blacklozenge… \\ $$$$ \\ $$$$\mathrm{Evaluate}:: \\ $$$$ \\ $$$$\Omega\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{sec}\theta}{\:\sqrt{\mathrm{4tan}^{\mathrm{2}} \theta+\mathrm{5}}}\mathrm{d}\theta \\ $$$$ \\ $$$$…\spadesuit\boldsymbol{\mathrm{L}\phi\mathrm{rD}}\:\boldsymbol{\varnothing\mathrm{sE}}\spadesuit… \\ $$$$ \\ $$$$…\clubsuit\boldsymbol{\mathrm{GooD}}\:\boldsymbol{\mathrm{LucK}}\clubsuit \\ $$
Answered by MJS_new last updated on 18/Oct/20
∫((sec θ)/( (√(5+4tan^2  θ))))dθ=  =sign (cos x)((√5)/5)∫(dθ/( (√(1−(1/5)sin^2  θ))))=  =sign (cos x)((√5)/5)F (x∣(1/5)) +C  ⇒ Ω is not defined  [plot ((sec θ)/( (√(5+4tan^2  θ)))) and you′ll see why]
$$\int\frac{\mathrm{sec}\:\theta}{\:\sqrt{\mathrm{5}+\mathrm{4tan}^{\mathrm{2}} \:\theta}}{d}\theta= \\ $$$$=\mathrm{sign}\:\left(\mathrm{cos}\:{x}\right)\frac{\sqrt{\mathrm{5}}}{\mathrm{5}}\int\frac{{d}\theta}{\:\sqrt{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{5}}\mathrm{sin}^{\mathrm{2}} \:\theta}}= \\ $$$$=\mathrm{sign}\:\left(\mathrm{cos}\:{x}\right)\frac{\sqrt{\mathrm{5}}}{\mathrm{5}}\mathrm{F}\:\left({x}\mid\frac{\mathrm{1}}{\mathrm{5}}\right)\:+{C} \\ $$$$\Rightarrow\:\Omega\:\mathrm{is}\:\mathrm{not}\:\mathrm{defined} \\ $$$$\left[\mathrm{plot}\:\frac{\mathrm{sec}\:\theta}{\:\sqrt{\mathrm{5}+\mathrm{4tan}^{\mathrm{2}} \:\theta}}\:\mathrm{and}\:\mathrm{you}'\mathrm{ll}\:\mathrm{see}\:\mathrm{why}\right] \\ $$

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