What is the integral from-pi to pi of pisin^2(x) ?

The integral from-pi to pi of pisin^2(x) is pi^2

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integral from-pi to pi of pisin^2(x)

\int_{- π}^{π} π sin^{2} (x) d x

π^{2}

Decimal Notation

9.86960 \dots

Solution steps

\int_{- π}^{π} π sin^{2} (x) d x

Take the constant out:

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

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

Rewrite using trig identities

= π \cdot \int_{- π}^{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

= π \frac{1}{2} \cdot \int_{- π}^{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

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

\int_{- π}^{π} 1 d x = 2 π

\int_{- π}^{π} cos (2 x) d x = 0

= π \frac{1}{2} (2 π - 0)

Simplify

π \frac{1}{2} (2 π - 0) : π^{2}

= π^{2}

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What is the integral from-pi to pi of pisin^2(x) ?
The integral from-pi to pi of pisin^2(x) is pi^2