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

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

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

\int_{0}^{π} sin^{2} (2 x) d x

\frac{π}{2}

Decimal Notation

1.57079 \dots

Solution steps

\int_{0}^{π} sin^{2} (2 x) d x

Rewrite using trig identities

= \int_{0}^{π} \frac{1}{2} (1 - cos (4 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 (4 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 (4 x) d x)

\int_{0}^{π} 1 d x = π

\int_{0}^{π} cos (4 x) d x = 0

= \frac{1}{2} (π - 0)

Simplify

\frac{1}{2} (π - 0) : \frac{π}{2}

= \frac{π}{2}

x \to 4 lim (3 - x)

y^{^{'}} = \frac{x ^{4}}{y}

y^{^{'}} - \frac{1}{2} y = 2 cos (x)

x \to 1 lim (\frac{4}{2 x - 2})

\frac{\partial}{\partial x} (6 x^{2} y^{2} + 5 x + 8 y^{3})

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