The value of _0^20 ( ^4x+ ^4x)dx is equal to:

Question

The value of $\int _{0}^{20\pi }({\sin }^{4}x+{\cos }^{4}x)dx$ is equal to:

Accepted Answer

$I=\int _{0}^{20\pi }(1-2{\sin }^{2}x{\cos }^{2}x)\cdot dx$

$=20\pi -\frac{1}{2}\int _{0}^{20\pi }{\sin }^{2}2x\, dx$

$=20\pi -\frac{1}{2}(40)\int _{0}^{\pi /2}{\sin }^{2}2x$

$=20\pi -\frac{20}{2}{(x-\frac{\sin 2x}{2})}_{\pi /2}^{0}$

$=20\pi -10(\frac{\pi }{2})=15\pi $

Solutions · 1