∫$$0$$π$$/2$$ $$sin3/2$$⁡$$xsin3/2$$⁡$$x+cos3/2$$⁡xdx is equal to

Question

∫$$0$$π$$/2$$ $$sin3/2$$⁡$$xsin3/2$$⁡$$x+cos3/2$$⁡xdx is equal to

Infinity Learn · Accepted Answer

Let,$$I=$$∫$$0$$π$$/2$$ $$sin3/2$$⁡$$xsin3/2$$⁡$$x+cos3/2$$⁡$$xdx----ithen$$  $$I=$$∫$$0$$π$$/2$$ $$sin3/2$$⁡π$$2$$−$$xsin3/2$$⁡π$$2$$−$$x+cos3/2$$⁡π$$2$$−xdx ∵∫$$0a$$ $$f(x)dx=$$∫$$0a$$ $$f(a$$−$$x)dx$$⇒$$I=$$∫$$0$$π$$/2$$ $$cos3/2$$⁡$$xcos3/2$$⁡$$x+sin3/2$$⁡$$xdx-----ii$$∵$$sin$$⁡π$$2$$−$$x=$$$$cos$$⁡x and $$cos$$⁡π$$2$$−$$x=$$$$sin$$⁡xOn adding Eqs. (i) and $$(iil$$, we $$get2I=$$∫$$0$$π$$/2$$ $$sin3/2$$⁡$$x+cos3/2$$⁡$$xsin3/2$$⁡$$x+cos3/2$$⁡$$xdx=$$∫$$0$$π$$/2$$ $$1dx=$$[x]$$0$$π$$/2=$$π$$2$$−$$0$$⇒$$I=$$π$$4$$

$\int_{0}^{π / 2} \frac{\sin^{3 / 2} x}{\sin^{3 / 2} x + \cos^{3 / 2} x} dx$ is equal to

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∫0π/2 sin3/2⁡xsin3/2⁡x+cos3/2⁡xdx is equal to

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$\int_{0}^{π / 2} \frac{\sin^{3 / 2} x}{\sin^{3 / 2} x + \cos^{3 / 2} x} dx$ is equal to