Question Number 216281 by MrGaster last updated on 02/Feb/25
$$\mathrm{Prove}:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {d}\phi\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left(\mathrm{sin}\theta\:\mathrm{cos}\:\theta\right)\mathrm{sin}\theta\:{d}\theta=\frac{\pi}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$
Commented by mr W last updated on 03/Feb/25
![it′s wrong! example: f(x)=x+1 f(sin θ cos θ)=sin θ cos θ+1 ∫_0 ^(π/2) f(sin θ cos θ) sin θ dθ =∫_0 ^(π/2) (sin θ+sin^2 θ cos θ) dθ =[−cos θ+((sin^3 θ)/3)]_0 ^(π/2) =1+(1/3)=(4/3) but ∫_0 ^1 f(x)dx =∫_0 ^1 (x+1)dx=[x+(x^2 /2)]_0 ^1 =(3/2)≠(4/3)](Q216314.png)
$${it}'{s}\:{wrong}! \\ $$$${example}:\:{f}\left({x}\right)={x}+\mathrm{1} \\ $$$${f}\left(\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta\right)=\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta+\mathrm{1} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left(\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta\right)\:\mathrm{sin}\:\theta\:{d}\theta \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{sin}\:\theta+\mathrm{sin}^{\mathrm{2}} \:\theta\:\mathrm{cos}\:\theta\right)\:{d}\theta \\ $$$$=\left[−\mathrm{cos}\:\theta+\frac{\mathrm{sin}^{\mathrm{3}} \:\theta}{\mathrm{3}}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \\ $$$$=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}=\frac{\mathrm{4}}{\mathrm{3}} \\ $$$${but} \\ $$$$\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \left({x}+\mathrm{1}\right){dx}=\left[{x}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right]_{\mathrm{0}} ^{\mathrm{1}} =\frac{\mathrm{3}}{\mathrm{2}}\neq\frac{\mathrm{4}}{\mathrm{3}} \\ $$
Commented by MrGaster last updated on 04/Feb/25
$$\mathrm{I}\:\mathrm{think}\:\mathrm{so}\:\mathrm{but}\:\mathrm{a}\:\mathrm{friend}\:\mathrm{of}\:\mathrm{mine}\:\mathrm{onces} \\ $$$$\mathrm{ent}\:\mathrm{me}\:\mathrm{a}\:\mathrm{proof}\:\mathrm{of}\:\mathrm{this}\:\mathrm{proposition}.\mathrm{T} \\ $$$$\mathrm{he}\:\mathrm{picture}\:\mathrm{below}\:\mathrm{is}\:\mathrm{the}\:\mathrm{proof}. \\ $$
Commented by MrGaster last updated on 04/Feb/25
Commented by mr W last updated on 04/Feb/25
![it has too many errors even with simple elementary things! e.g. ∫_0 ^(π/2) f(((sin 2θ)/2))sin θ dθ≠f(((sin 2θ)/2))[(−cos θ)]_0 ^(π/2)](Q216322.png)
$${it}\:{has}\:{too}\:{many}\:{errors}\:{even}\:{with}\: \\ $$$${simple}\:{elementary}\:{things}!\:{e}.{g}. \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left(\frac{\mathrm{sin}\:\mathrm{2}\theta}{\mathrm{2}}\right)\mathrm{sin}\:\theta\:{d}\theta\neq{f}\left(\frac{\mathrm{sin}\:\mathrm{2}\theta}{\mathrm{2}}\right)\left[\left(−\mathrm{cos}\:\theta\right)\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \\ $$