The value of ∫$$02$$π $$cos2n$$⁡$$xcos2n$$⁡$$x+sin2n$$⁡xdx is

Question

The value of ∫$$02$$π $$cos2n$$⁡$$xcos2n$$⁡$$x+sin2n$$⁡xdx is

Infinity Learn · Accepted Answer

Here $$f(x)=cos2n$$⁡$$xcos2n$$⁡$$x+sin2n$$⁡x for which $$f(x)=f(2$$π−$$x)$$⇒ $$I=$$∫$$02$$π $$cos2n$$⁡$$xcos2n$$⁡$$x+sin2n$$⁡$$xdx=2$$∫$$0$$π $$cos2n$$⁡$$xcos2n$$⁡$$x+sin2n$$⁡xdx Again, $$f(x)=f($$π−$$x)$$⇒ $$I=4$$∫$$0$$π$$/2$$ $$cos2n$$⁡$$xcos2n$$⁡$$x+sin2n$$⁡$$xdx---(1)=4$$∫$$0$$π$$/2$$ $$cos2n$$⁡π$$2$$−$$xcos2n$$⁡π$$2$$−$$x+sin2n$$⁡π$$2$$−xdx (by$$ property $$IV) $$=4$$∫$$0$$π$$/2$$ $$sin2n$$⁡$$xsin2n$$⁡$$x+cos2n$$⁡$$xdx-------(2)$$ Adding (1) and (2), we have $$2I=4$$∫$$0$$π$$/2$$ $$1dx$$⇒$$I=$$π

$The value of \int_{0}^{2 π} \frac{\cos^{2 n} x}{\cos^{2 n} x + \sin^{2 n} x} d x is$

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The value of ∫02π cos2n⁡xcos2n⁡x+sin2n⁡xdx is

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$The value of \int_{0}^{2 π} \frac{\cos^{2 n} x}{\cos^{2 n} x + \sin^{2 n} x} d x is$