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Question Number 216281 by MrGaster last updated on 02/Feb/25

$$\mathrm{Prove}:{\int }_0^{\frac{\pi }{2}}d\varphi {\int }_0^{\frac{\pi }{2}}f\left(\sin \theta \cos \theta \right)\sin \theta d\theta =\frac{\pi }{2}{\int }_0^{1}f\left(x\right)dx$$

Commented by mr W last updated on 03/Feb/25

$${it}^{\prime }swrong!$$$$example:f\left(x\right)=x+1$$$$f\left(\sin \theta \cos \theta \right)=\sin \theta \cos \theta +1$$$${\int }_0^{\frac{\pi }{2}}f\left(\sin \theta \cos \theta \right)\sin \theta d\theta$$$$={\int }_0^{\frac{\pi }{2}}(\sin \theta +{\sin }^2\theta \cos \theta )d\theta$$$$={[-\cos \theta +\frac{{\sin }^3\theta }{3}]}_0^{\frac{\pi }{2}}$$$$=1+\frac{1}{3}=\frac{4}{3}$$$$but$$$${\int }_0^{1}f\left(x\right)dx$$$$={\int }_0^1(x+1)dx={[x+\frac{x^2}{2}]}_0^1=\frac{3}{2}\ne \frac{4}{3}$$

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

$$ithastoomanyerrorsevenwith$$$$simpleelementarythings!e.g.$$$${\int }_0^{\frac{\pi }{2}}f\left(\frac{\sin 2\theta }{2}\right)\sin \theta d\theta \ne f\left(\frac{\sin 2\theta }{2}\right){\left[(-\cos \theta )\right]}_0^{\frac{\pi }{2}}$$

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