∫$$0$$π$$/2$$ $$sin$$⁡xsin⁡$$x+$$$$cos$$⁡xdx is equal to

Question

∫$$0$$π$$/2$$ $$sin$$⁡xsin⁡$$x+$$$$cos$$⁡xdx is equal to

Infinity Learn · Accepted Answer

Firstly, reduce the integrand into simplest form by usingthe property ∫$$0a$$ $$f(x)dx=$$∫$$0a$$ $$f(a$$−$$x)dx$$ add them and integrate.$$LetI=$$∫$$0$$π$$/2$$ $$sin$$⁡xsin⁡$$x+$$$$cos$$⁡$$xdx---ithen$$    $$I=$$∫$$0$$π$$/2$$ $$sin$$⁡π$$2$$−xsin⁡π$$2$$−$$x+$$$$cos$$⁡π$$2$$−xdx      ∵∫$$0a$$ $$f(x)dx=$$∫$$0a$$ $$f(a$$−$$x)dx$$⇒$$I=$$∫$$0$$π$$/2$$ $$cos$$⁡xcos⁡$$x+$$$$sin$$⁡$$xdx-----ii$$∵$$sin$$⁡π$$2$$−$$x=$$$$cos$$⁡x and $$cos$$⁡π$$2$$−$$x=$$$$sin$$⁡xOn adding Eqs. (i) and (ii), we $$get2l=$$∫$$0$$π$$/2$$ $$sin$$⁡$$x+$$$$cos$$⁡xsin⁡$$x+$$$$cos$$⁡$$xdx=$$∫$$0$$π$$/2$$ $$1dx=$$[x]$$0$$π$$/2=$$π$$2$$−$$0$$⇒ $$I=$$π$$4$$

$\int_{0}^{π / 2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$ is equal to

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∫0π/2 sin⁡xsin⁡x+cos⁡xdx is equal to

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$\int_{0}^{π / 2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$ is equal to