Let $$f(x)=$$$$sin$$⁡$$x+$$∑$$r=02$$ $$cos$$⁡$$x+2$$ r$$π$$$$3$$ $$g(x)=$$$$cos$$⁡$$x+$$∑$$r=02$$ $$sin$$⁡$$x+2$$ r$$π$$$$3$$ $$h(x)=max{f(x)$$,$$g(x)}$$∫−ππ $$g{f(x)}$$ $$sin$$ $$2x$$ dx is equalent to∫π$$/4$$π [ $$h(x)$$ ] $${f(x)}2dx$$ where [.] denotes greatest integer function is∫$$0$$π $${g(x)}337dx=$$

Question

Let $$f(x)=$$$$sin$$⁡$$x+$$∑$$r=02$$ $$cos$$⁡$$x+2$$ r$$π$$$$3$$ $$g(x)=$$$$cos$$⁡$$x+$$∑$$r=02$$ $$sin$$⁡$$x+2$$ r$$π$$$$3$$ $$h(x)=max{f(x)$$,$$g(x)}$$∫−ππ $$g{f(x)}$$ $$sin$$ $$2x$$ dx  is equalent to∫π$$/4$$π [ $$h(x)$$ ] $${f(x)}2dx$$ where [.] denotes greatest integer function is∫$$0$$π $${g(x)}337dx=$$

Infinity Learn · Accepted Answer

$$f(x)=$$$$sin$$⁡$$x+$$∑$$r=02$$ $$cos$$⁡$$x+2r$$$$π$$$$3=$$$$sin$$⁡$$x+$$$$cos$$⁡$$x+$$$$cos$$⁡$$x+2$$$$π$$$$3+$$$$cos$$⁡$$x+4$$$$π$$$$3=$$$$sin$$⁡$$x+$$$$cos$$⁡x−$$2$$⋅$$12cos$$⁡$$x=$$$$sin$$⁡xsimilarly $$g(x)$$ $$=$$ cosx∫−ππ g $${f(x)}$$ $$sin2x$$ $$dx=$$∫−ππ $$cos$$(sinx)⋅$$sin2x$$ $$dx=0$$ since integrand is odd∫π$$/4$$π [$$h(x)$$]$${f(x)}2dx=$$∫$$x/4$$π [$$sin$$⁡x]⋅$$sin2$$⁡$$xdx=0$$∫$$0$$π $${g(x)}337dx=$$∫$$0$$π $$cos337$$⁡$$xdx=$$∫$$0$$π $$cos337$$⁡$$($$π−$$x)dx=$$−∫$$0$$π $$cos337$$⁡xdx ∴∫$$0$$π $$cos337$$⁡$$xdx=0$$

Let f(x)=sin⁡x+∑r=02 cos⁡x+2 rπ3 g(x)=cos⁡x+∑r=02 sin⁡x+2 rπ3 h(x)=max{f(x),g(x)}∫−ππ g{f(x)} sin 2x dx is equalent to∫π/4π [ h(x) ] {f(x)}2dx where [.] denotes greatest integer function is∫0π {g(x)}337dx=

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