What is the integral from 0 to pi of 1/2 sin^2(x) ?

The integral from 0 to pi of 1/2 sin^2(x) is pi/4

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integral from 0 to pi of 1/2 sin^2(x)

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

\frac{π}{4}

Decimal

0.78539 \dots

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

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

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

\int_{0}^{π} 1 d x = π

\int_{0}^{π} cos (2 x) d x = 0

= \frac{1}{2} \cdot \frac{1}{2} (π - 0)

Simplify

\frac{1}{2} \cdot \frac{1}{2} (π - 0) : \frac{π}{4}

= \frac{π}{4}

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What is the integral from 0 to pi of 1/2 sin^2(x) ?
The integral from 0 to pi of 1/2 sin^2(x) is pi/4