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Question Number 40151 by maxmathsup by imad last updated on 16/Jul/18

let F(x) = ∫_0 ^(π/2) cos(xsint)dt 1) prove that ∀u ∈R 1−(u^2 /2) ≤cosu≤1−(u^2 /2) +(u^4 /(24)) 2) prove that (π/2)(1−(x^2 /4))≤F(x)≤ (π/2)(1−(x^2 /4) +(x^4 /(64)))

$${let}\:{F}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}\left({xsint}\right){dt} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\forall{u}\:\in{R}\:\:\mathrm{1}−\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:\leqslant{cosu}\leqslant\mathrm{1}−\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:+\frac{{u}^{\mathrm{4}} }{\mathrm{24}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\right)\leqslant{F}\left({x}\right)\leqslant\:\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\:+\frac{{x}^{\mathrm{4}} }{\mathrm{64}}\right) \\ $$

Commented by math khazana by abdo last updated on 19/Jul/18

1) we have cos(u)=Σ_(n=0) ^∞ (((−1)^n u^(2n) )/((2n)!)) =1−(u^2 /2) +(u^4 /((4)!)) −(u^6 /(6!)) +.... ∀u∈R ⇒ 1−(u^2 /2) ≤cosu ≤1−(u^2 /2) +(u^4 /(24)) 2) we get 1−(((xsint)^2 )/2) ≤cos(xsint)≤1−(((xsint)^2 )/2) +(((xsint)^4 )/(24)) ⇒ ∫_0 ^(π/2) (1−(x^2 /2)sin^2 t)dt ≤ ∫_0 ^(π/2) cos(xsint)dt ≤ ∫_0 ^(π/2) (1−((x^2 sin^2 t)/2) +((x^4 sin^4 )/(24)))dt but ∫_0 ^(π/2) (1−(x^2 /2)sin^2 t)dt =(π/2) −(x^2 /2) ∫_0 ^(π/2) ((1−cos(2t))/2)dt =(π/2) −(x^2 /4) (π/2) =(π/2)(1−(x^2 /4)) also ∫_0 ^(π/2) (1−((x^2 sin^2 t)/2) +((x^4 sin^4 t)/(24)))dt =(π/2)(1−(x^2 /4)) +(x^4 /(24)) ∫_0 ^(π/2) (((1−cos(2t))/2))^2 dt =(π/2)(1−(x^2 /4)) +(x^4 /(96)) ∫_0 ^(π/2) (1−2cos(2t) +cos^2 (2t))dt =(π/2)(1−(x^2 /2)) +(x^4 /(96)) (π/2) + (x^4 /(96)) ∫_0 ^(π/2) ((1+cos(4t))/2)dt =(π/2)(1−(x^2 /2))+(π/2)( (1/(96)) +(1/(2.96)))x^4 =(π/2)(1−(x^2 /2)) +(x^4 /(64)).(π/2) =(π/2)(1−(x^2 /2) +(x^4 /(64))) ⇒ tbe result is proved.

$$\left.\mathrm{1}\right)\:{we}\:{have}\:\:{cos}\left({u}\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} {u}^{\mathrm{2}{n}} }{\left(\mathrm{2}{n}\right)!} \\ $$$$=\mathrm{1}−\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:\:+\frac{{u}^{\mathrm{4}} }{\left(\mathrm{4}\right)!}\:−\frac{{u}^{\mathrm{6}} }{\mathrm{6}!}\:+....\:\forall{u}\in{R}\:\Rightarrow \\ $$$$\mathrm{1}−\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:\leqslant{cosu}\:\leqslant\mathrm{1}−\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:+\frac{{u}^{\mathrm{4}} }{\mathrm{24}} \\ $$$$\left.\mathrm{2}\right)\:{we}\:{get}\:\mathrm{1}−\frac{\left({xsint}\right)^{\mathrm{2}} }{\mathrm{2}}\:\leqslant{cos}\left({xsint}\right)\leqslant\mathrm{1}−\frac{\left({xsint}\right)^{\mathrm{2}} }{\mathrm{2}} \\ $$$$+\frac{\left({xsint}\right)^{\mathrm{4}} }{\mathrm{24}}\:\Rightarrow \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}{sin}^{\mathrm{2}} {t}\right){dt}\:\leqslant\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}\left({xsint}\right){dt} \\ $$$$\leqslant\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} \:{sin}^{\mathrm{2}} {t}}{\mathrm{2}}\:+\frac{{x}^{\mathrm{4}} \:{sin}^{\mathrm{4}} }{\mathrm{24}}\right){dt}\:{but} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}{sin}^{\mathrm{2}} {t}\right){dt}\:=\frac{\pi}{\mathrm{2}}\:−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{1}−{cos}\left(\mathrm{2}{t}\right)}{\mathrm{2}}{dt} \\ $$$$=\frac{\pi}{\mathrm{2}}\:−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\:\frac{\pi}{\mathrm{2}}\:=\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\right)\:{also} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{1}−\frac{{x}^{\mathrm{2}} \:{sin}^{\mathrm{2}} {t}}{\mathrm{2}}\:+\frac{{x}^{\mathrm{4}} {sin}^{\mathrm{4}} {t}}{\mathrm{24}}\right){dt} \\ $$$$=\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\right)\:+\frac{{x}^{\mathrm{4}} }{\mathrm{24}}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\frac{\mathrm{1}−{cos}\left(\mathrm{2}{t}\right)}{\mathrm{2}}\right)^{\mathrm{2}} \:{dt} \\ $$$$=\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\right)\:+\frac{{x}^{\mathrm{4}} }{\mathrm{96}}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{1}−\mathrm{2}{cos}\left(\mathrm{2}{t}\right)\:+{cos}^{\mathrm{2}} \left(\mathrm{2}{t}\right)\right){dt} \\ $$$$=\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)\:+\frac{{x}^{\mathrm{4}} }{\mathrm{96}}\:\frac{\pi}{\mathrm{2}}\:+\:\frac{{x}^{\mathrm{4}} }{\mathrm{96}}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{1}+{cos}\left(\mathrm{4}{t}\right)}{\mathrm{2}}{dt} \\ $$$$=\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)+\frac{\pi}{\mathrm{2}}\left(\:\:\frac{\mathrm{1}}{\mathrm{96}}\:\:+\frac{\mathrm{1}}{\mathrm{2}.\mathrm{96}}\right){x}^{\mathrm{4}} \\ $$$$=\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)\:+\frac{{x}^{\mathrm{4}} }{\mathrm{64}}.\frac{\pi}{\mathrm{2}}\:=\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\:+\frac{{x}^{\mathrm{4}} }{\mathrm{64}}\right)\:\Rightarrow \\ $$$${tbe}\:{result}\:{is}\:{proved}. \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 17/Jul/18

p+iq=∫_0 ^(Π/2) e^(ixsint) dt ∫_0 ^(Π/2) 1+ixsint+((i^2 x^2 sin^2 t)/(2!))+((i^3 x^3 sin^3 t)/(3!))+((i^4 x^4 sin^4 t)/(4!))+.. p=F(x)= ∫_0 ^(Π/2) 1−((x^2 sin^2 t)/4)+((x^4 sin^4 t)/(24))+..dt 1≥sin^2 t≥0 ... 1≥sin^(2k) t≥0 2k=even so ∫_0 ^(Π/2) 1−(x^2 /4)dt≤F(x)≤∫_0 ^(Π/2) 1−(x^2 /4)+(x^4 /(24))dt (1−(x^2 /4))(Π/2)≤F(x)≤(1−(x^2 /4)+(x^4 /(24)))(Π/2)

$${p}+{iq}=\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} {e}^{{ixsint}} {dt} \\ $$$$\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \mathrm{1}+{ixsint}+\frac{{i}^{\mathrm{2}} {x}^{\mathrm{2}} {sin}^{\mathrm{2}} {t}}{\mathrm{2}!}+\frac{{i}^{\mathrm{3}} {x}^{\mathrm{3}} {sin}^{\mathrm{3}} {t}}{\mathrm{3}!}+\frac{{i}^{\mathrm{4}} {x}^{\mathrm{4}} {sin}^{\mathrm{4}} {t}}{\mathrm{4}!}+.. \\ $$$${p}={F}\left({x}\right)= \\ $$$$\:\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \mathrm{1}−\frac{{x}^{\mathrm{2}} {sin}^{\mathrm{2}} {t}}{\mathrm{4}}+\frac{{x}^{\mathrm{4}} {sin}^{\mathrm{4}} {t}}{\mathrm{24}}+..{dt} \\ $$$$\:\mathrm{1}\geqslant{sin}^{\mathrm{2}} {t}\geqslant\mathrm{0}\:\:...\:\:\mathrm{1}\geqslant{sin}^{\mathrm{2}{k}} {t}\geqslant\mathrm{0}\:\:\mathrm{2}{k}={even} \\ $$$${so}\:\:\:\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}{dt}\leqslant{F}\left({x}\right)\leqslant\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}+\frac{{x}^{\mathrm{4}} }{\mathrm{24}}{dt} \\ $$$$\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\right)\frac{\Pi}{\mathrm{2}}\leqslant{F}\left({x}\right)\leqslant\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}+\frac{{x}^{\mathrm{4}} }{\mathrm{24}}\right)\frac{\Pi}{\mathrm{2}} \\ $$$$ \\ $$

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