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 $$cos337$$⁡$$xdx=0$$

Evaluation of definite integrals

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Let $f (x) = \sin x + \sum_{r = 0}^{2} \cos (x + \frac{2 r π}{3})$
$g (x) = \cos x + \sum_{r = 0}^{2} \sin (x + \frac{2 r π}{3})$
$h (x) = max {f (x), g (x)}$

$\int_{- π}^{π} g {f (x)} \sin 2 x dx$ is equalent to

$\int_{π / 4}^{π} [h (x)] {f (x)}^{2} dx$ where [.] denotes greatest integer function is

$\int_{π / 4}^{π} [h (x)] {f (x)}^{2} dx = \int_{x / 4}^{π} [\sin x] \cdot \sin^{2} xdx = 0$

$\int_{0}^{π} {g (x)}^{337} dx =$

$\begin{aligned} \int_{0}^{π} {g (x)}^{337} dx = \int_{0}^{π} \cos^{337} xdx = \int_{0}^{π} \cos^{337} (π - x) dx \\ = - \int_{0}^{π} \cos^{337} xdx ∴ \int_{0}^{π} \cos^{337} xdx = 0 \end{aligned}$

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Evaluation of definite integrals

Unlock the full solution & master the concept.

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

f(x)=sin⁡x+∑r=02 cos⁡x+2rπ3=sin⁡x+cos⁡x+cos⁡x+2π3+cos⁡x+4π3=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/4π [ h(x) ] {f(x)}2dx where [.] denotes greatest integer function is

∫π/4π [h(x)]{f(x)}2dx=∫x/4π [sin⁡x]⋅sin2⁡xdx=0

∫0π {g(x)}337dx=

∫0π {g(x)}337dx=∫0π cos337⁡xdx=∫0π cos337⁡(π−x)dx=−∫0π cos337⁡xdx ∴∫0π cos337⁡xdx=0