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

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

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

\int_{- 2 π}^{2 π} π sin^{2} (x) d x

2 π^{2}

Decimal

19.73920 \dots

Solution steps

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

Rewrite using trig identities

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

\int_{- 2 π}^{2 π} 1 d x = 4 π

\int_{- 2 π}^{2 π} cos (2 x) d x = 0

= π \frac{1}{2} (4 π - 0)

Simplify

π \frac{1}{2} (4 π - 0) : 2 π^{2}

= 2 π^{2}

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