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Question Number 195846 by mnjuly1970 last updated on 11/Aug/23

{ (( Ω_1 = ∫_0 ^( (π/2)) (( x^( 2) )/(sin^( 2) (x))) dx )),(( ⇒ (Ω_1 /Ω_( 2) ) = ? )),(( Ω_( 2) = ∫_0 ^( (π/2)) (x/(tan(x))) dx)) :}

$$ \\ $$$$\:\:\:\:\:\:\begin{cases}{\:\:\:\Omega_{\mathrm{1}} \:=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\:{x}^{\:\mathrm{2}} }{{sin}^{\:\mathrm{2}} \left({x}\right)}\:{dx}\:}\\{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:\:\frac{\Omega_{\mathrm{1}} }{\Omega_{\:\mathrm{2}} }\:=\:?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:}\\{\:\:\Omega_{\:\mathrm{2}} =\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{x}}{{tan}\left({x}\right)}\:{dx}}\end{cases} \\ $$$$ \\ $$

Answered by sniper237 last updated on 11/Aug/23

by parts integration Ω_2 =(1/2)Ω_1

$${by}\:{parts}\:{integration}\:\Omega_{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{2}}\Omega_{\mathrm{1}} \\ $$

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