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

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

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

\int_{0}^{π} π (sin^{2} (x)) d x

\frac{π ^{2}}{2}

Decimal Notation

4.93480 \dots

Solution steps

\int_{0}^{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_{0}^{π} sin^{2} (x) d x

Rewrite using trig identities

= π \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

= π \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

= π \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

= π \frac{1}{2} (π - 0)

Simplify

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

= \frac{π ^{2}}{2}

x \to 5 - lim (x)

\int sin (2) x^{2} d x

\frac{d}{d x} (8 π x^{2})

f (x) = sin (x) cos (x) - x^{2}

\int x e^{- \frac{2}{5} x} d x

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