Altoinferno Altoinferno
  • 04-06-2019
  • Mathematics
contestada

What is the antiderivative from 0 to Pi/3 of (Cos(4x)+Sin(4x))^2​

What is the antiderivative from 0 to Pi3 of Cos4xSin4x2 class=

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LammettHash
LammettHash LammettHash
  • 04-06-2019

The quickest method would probably be to just expand the integrand:

[tex](\cos4x+\sin4x)^2=\cos^24x+2\cos4x\sin4x+\sin^24x=1+\sin8x[/tex]

Then

[tex]\displaystyle\int_0^{\pi/3}(1+\sin8x)\,\mathrm dx=\left(x-\dfrac18\cos8x\right)\bigg|_0^{\pi/3}[/tex]

[tex]=\left(\dfrac\pi3-\dfrac18\cos\dfrac{8\pi}3\right)-\left(0-\dfrac18\right)=\boxed{\dfrac3{16}+\dfrac\pi3}[/tex]

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