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Question Number 17210 by Arnab Maiti last updated on 02/Jul/17

$$\mathrm{prove}\mathrm{that}{\int }_0^{\mathrm{\Pi }}f\left(\sin \mathrm{x}\right)\mathrm{dx}=2\times {\int }_0^{\frac{\mathrm{\Pi }}{2}}f\left(\sin \mathrm{x}\right)\mathrm{dx}$$

Answered by mrW1 last updated on 02/Jul/17

$$\mathrm{let}\mathrm{I}={\int }_{\frac{\pi }{2}}^{\pi }\mathrm{f}\left(\sin \mathrm{x}\right)\mathrm{dx}$$$$\mathrm{t}=\pi -\mathrm{x}$$$$\mathrm{x}=\pi -\mathrm{t}$$$$\sin \mathrm{x}=\sin (\pi -\mathrm{t})=\sin \mathrm{t}$$$$\mathrm{dx}=-\mathrm{dt}$$$$\mathrm{I}={\int }_{\frac{\pi }{2}}^{\pi }\mathrm{f}\left(\sin \mathrm{x}\right)\mathrm{dx}={\int }_{\frac{\pi }{2}}^0\mathrm{f}\left(\sin \mathrm{t}\right)(-\mathrm{dt})$$$$=-{\int }_{\frac{\pi }{2}}^0\mathrm{f}\left(\sin \mathrm{t}\right)\mathrm{dt}$$$$={\int }_0^{\frac{\pi }{2}}\mathrm{f}\left(\sin \mathrm{t}\right)\mathrm{dt}$$$$={\int }_0^{\frac{\pi }{2}}\mathrm{f}\left(\sin \mathrm{x}\right)\mathrm{dx}$$$$$$$$\Rightarrow {\int }_0^{\pi }\mathrm{f}\left(\sin \mathrm{x}\right)\mathrm{dx}={\int }_0^{\frac{\pi }{2}}\mathrm{f}\left(\sin \mathrm{x}\right)\mathrm{dx}+{\int }_{\frac{\pi }{2}}^{\pi }\mathrm{f}\left(\sin \mathrm{x}\right)\mathrm{dx}$$$$={\int }_0^{\frac{\pi }{2}}\mathrm{f}\left(\sin \mathrm{x}\right)\mathrm{dx}+{\int }_0^{\frac{\pi }{2}}\mathrm{f}\left(\sin \mathrm{x}\right)\mathrm{dx}$$$$=2{\int }_0^{\frac{\pi }{2}}\mathrm{f}\left(\sin \mathrm{x}\right)\mathrm{dx}$$

Commented by Arnab Maiti last updated on 02/Jul/17

$$\mathrm{Thank}\mathrm{u}\mathrm{sir}.\mathrm{I}\mathrm{really}\mathrm{appriciate}.$$

Commented by Arnab Maiti last updated on 02/Jul/17

$$\mathrm{Sir}\mathrm{can}\mathrm{you}\mathrm{give}\mathrm{me}\mathrm{some}\mathrm{more}\mathrm{questions}$$$$\mathrm{like}\mathrm{that}?\mathrm{Please}\mathrm{do}.$$

Commented by mrW1 last updated on 02/Jul/17

$$\mathrm{But}\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{have}\mathrm{any}\mathrm{such}\mathrm{questions}.$$

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