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

The integral from 0 to pi of sin^2(x/3) is (4pi-3sqrt(3))/8

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integral from 0 to pi of sin^2(x/3)

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

\frac{4 π - 3 3}{8}

Decimal

0.92127 \dots

Solution steps

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

Rewrite using trig identities

= \int_{0}^{π} \frac{1}{2} (1 - cos (\frac{2 x}{3})) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{0}^{π} 1 - cos (\frac{2 x}{3}) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int_{0}^{π} 1 d x - \int_{0}^{π} cos (\frac{2 x}{3}) d x)

\int_{0}^{π} 1 d x = π

\int_{0}^{π} cos (\frac{2 x}{3}) d x = \frac{3 3}{4}

= \frac{1}{2} (π - \frac{3 3}{4})

Simplify

\frac{1}{2} (π - \frac{3 3}{4}) : \frac{4 π - 3 3}{8}

= \frac{4 π - 3 3}{8}

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What is the integral from 0 to pi of sin^2(x/3) ?
The integral from 0 to pi of sin^2(x/3) is (4pi-3sqrt(3))/8