The value of ∫−π$$/2$$π$$/2$$ xsin⁡$$xex+1dx$$ is equal to

Question

The value of ∫−π$$/2$$π$$/2$$ xsin⁡$$xex+1dx$$ is equal to

Infinity Learn · Accepted Answer

$$I=$$∫−π$$/2$$π$$/2$$ xsin⁡$$xex+1dx=$$∫−π$$/2$$π$$/2$$ $$($$−$$x)$$$$sin$$⁡$$($$−$$x)e$$−$$x+1dx$$ (Prop$$. $$11) $$=$$∫−π$$/2$$π$$/2$$ (xsin$$⁡$$x)exex+1dx$$   $$2I=$$∫−π$$/2$$π$$/2$$ xsin⁡$$xex+1+$$∫−π$$/2$$π$$/2$$ $$ex(xsin$$⁡$$x)ex+1dx$$   $$=$$∫−π$$/2$$π$$/2$$ $$ex+1(xsin$$⁡$$x)ex+1dx$$   $$=$$∫−π$$/2$$π$$/2$$ xsin⁡$$xdx=2$$∫$$0$$π$$/2$$ xsin⁡xdx   $$($$ Prop. $$12)     $$=2$$−xcos⁡$$x0$$π$$/2+$$∫$$0$$π$$/2$$ $$cos$$⁡xdx     $$=2sin$$⁡$$x0$$π$$/2=2$$⇒   $$I=1$$.

The value of $\int_{- π / 2}^{π / 2} \frac{x \sin x}{e^{x} + 1} d x$ is equal to

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The value of ∫−π/2π/2 xsin⁡xex+1dx is equal to

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The value of $\int_{- π / 2}^{π / 2} \frac{x \sin x}{e^{x} + 1} d x$ is equal to