Evaluate ∫$$0$$π xlog⁡$$sin$$⁡xdx

Question

Evaluate ∫$$0$$π xlog⁡$$sin$$⁡xdx

Infinity Learn · Accepted Answer

Let $$I=$$∫$$0$$π xlog⁡$$sin$$⁡$$xdx--------(1)I=$$∫$$0$$π $$($$π−$$x)log$$⁡$$sin$$⁡$$($$π−$$x)dx=$$∫$$0$$π $$($$π−$$x)log$$⁡$$sin$$⁡$$xdx---------(2)$$ Adding equations (1) and (2), we get $$2I=$$π∫$$0$$π log⁡$$sin$$⁡xdx or  $$2I=2$$π∫$$0$$π$$/2$$ log⁡$$sin$$⁡xdx $$I=$$π∫$$0$$π$$/2$$ log⁡$$sin$$⁡xdx $$----(3)$$ $$I=$$π∫$$0$$π$$/2$$ log⁡$$sin$$$$($$π$$2-$$⁡$$x)dx$$  $$=$$π∫$$0$$π$$/2$$ log⁡$$cos$$⁡xdx $$----(4)(3)$$ $$+$$ (4)2I=$$π∫$$0$$π$$/2$$ log⁡$$sin$$⁡$$x+logcosxdx$$    $$(take$$ $$logsinxcosx=log2sinxcosx2)=$$π∫$$0$$π$$/2$$ log⁡$$sin$$⁡$$2xdx-$$π∫$$0$$π$$/2$$ log⁡$$2dxput$$ $$2x=t$$,  $$dx=dt2$$  $$UL=$$π, $$LL=02I=$$π$$2$$∫$$0$$π log⁡$$sin$$⁡$$tdt-$$π$$log2$$$$π$$$$22I=$$π∫$$0$$$$π$$$$2$$ log⁡$$sin$$⁡$$tdt-$$π$$22log22I=I-$$π$$22log2I=-$$π$$22log2=$$π$$22log$$⁡$$(1/2)

$Evaluate \int_{0}^{π} x \log \sin x d x$

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Evaluate ∫0π xlog⁡sin⁡xdx

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$Evaluate \int_{0}^{π} x \log \sin x d x$