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

The integral from 0 to pi of |sin(2x)|sin(x) is 4/3

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

\int_{0}^{π} ∣ sin (2 x) ∣ sin (x) d x

\frac{4}{3}

Decimal

1.33333 \dots

Solution steps

\int_{0}^{π} ∣ sin (2 x) ∣ sin (x) d x

Eliminate Absolutes

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

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

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

= \frac{2}{3} + \frac{2}{3}

Simplify

\frac{2}{3} + \frac{2}{3} : \frac{4}{3}

= \frac{4}{3}

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What is the integral from 0 to pi of |sin(2x)|sin(x) ?
The integral from 0 to pi of |sin(2x)|sin(x) is 4/3