The value of ∫$$0$$π$$/2$$ $$sin$$⁡$$2$$θ$$sin$$⁡θdθ is

Question

The value of ∫$$0$$π$$/2$$ $$sin$$⁡$$2$$θ$$sin$$⁡θdθ is

Infinity Learn · Accepted Answer

$$I=$$∫$$0$$π$$/2$$ $$sin$$⁡$$2$$θ$$sin$$⁡θdθ$$=$$∫$$0$$π$$/2$$ $$sin$$⁡$$($$π−$$2$$θ$$)$$$$sin$$⁡$$($$π$$/2$$−θ$$)d$$θ$$=$$∫$$0$$π$$/2$$ $$sin$$⁡$$2$$θ$$cos$$⁡θdθ$$2I=$$∫$$0$$π$$/2$$ $$sin$$⁡$$2$$θ$$($$$$sin$$⁡θ$$+$$$$cos$$⁡θ$$)d$$θIn $$2I$$, put $$sin$$⁡θ−$$cos$$⁡θ$$=t$$, so that $$t2=1$$−$$sin$$⁡$$2$$θ⇒$$1$$−$$t2=$$$$sin$$⁡$$2$$θ. Also $$dt=($$$$cos$$⁡θ$$+$$$$sin$$⁡θ$$)d$$θ∴ $$2I=$$∫−$$11$$ $$1$$−$$t2dt$$∣$$=2t21$$−$$t2+12sin$$−$$1$$⁡$$t01=2$$$$π$$$$4=$$π$$2$$⇒ $$I=$$π$$4$$.

The value of $\int_{0}^{π / 2} \sqrt{\sin 2 θ} \sin θ d θ$ is

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The value of ∫0π/2 sin⁡2θsin⁡θdθ is

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The value of $\int_{0}^{π / 2} \sqrt{\sin 2 θ} \sin θ d θ$ is