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

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

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

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

\frac{π}{2}

Decimal

1.57079 \dots

Solution steps

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

Rewrite using trig identities

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

\int_{0}^{π} 1 d x = π

\int_{0}^{π} cos (x) d x = 0

= \frac{1}{2} (π - 0)

Simplify

\frac{1}{2} (π - 0) : \frac{π}{2}

= \frac{π}{2}

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