What is the integral from 0 to 2pi of |sin(2x)| ?

The integral from 0 to 2pi of |sin(2x)| is 4

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integral from 0 to 2pi of |sin(2x)|

\int_{0}^{2 π} ∣ sin (2 x) ∣ d x

4

Solution steps

\int_{0}^{2 π} ∣ sin (2 x) ∣ d x

Eliminate Absolutes

= \int_{0}^{\frac{π}{2}} sin (2 x) d x + \int_{\frac{π}{2}}^{π} - sin (2 x) d x + \int_{π}^{\frac{3 π}{2}} sin (2 x) d x + \int_{\frac{3 π}{2}}^{2 π} - sin (2 x) d x

\int_{0}^{\frac{π}{2}} sin (2 x) d x = 1

\int_{\frac{π}{2}}^{π} - sin (2 x) d x = 1

\int_{π}^{\frac{3 π}{2}} sin (2 x) d x = 1

\int_{\frac{3 π}{2}}^{2 π} - sin (2 x) d x = 1

= 1 + 1 + 1 + 1

Simplify

= 4

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\int_{0}^{54} x (\frac{1}{1458} x) d x

What is the integral from 0 to 2pi of |sin(2x)| ?
The integral from 0 to 2pi of |sin(2x)| is 4