What is the integral from 0 to 3pi of sin^2(x) ?

The integral from 0 to 3pi of sin^2(x) is (3pi)/2

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integral from 0 to 3pi of sin^2(x)

\int_{0}^{3 π} sin^{2} (x) d x

\frac{3 π}{2}

Decimal Notation

4.71238 \dots

Solution steps

\int_{0}^{3 π} sin^{2} (x) d x

Rewrite using trig identities

= \int_{0}^{3 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 \int_{0}^{3 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} (\int_{0}^{3 π} 1 d x - \int_{0}^{3 π} cos (2 x) d x)

\int_{0}^{3 π} 1 d x = 3 π

\int_{0}^{3 π} cos (2 x) d x = 0

= \frac{1}{2} (3 π - 0)

Simplify

\frac{1}{2} (3 π - 0) : \frac{3 π}{2}

= \frac{3 π}{2}

\int \frac{x}{1 + x ^{2}} d x

y^{^{''}} - 4 y^{^{'}} + 9 y = 0, y (0) = 0, y^{^{'}} (0) = - 8

f (x) = \frac{x ^{3}}{9}

\frac{d y}{d x} - 3 y = - 7 sin (x) - 7 cos (x)

x \to + 1 lim (e^{x - 1} - 1)

What is the integral from 0 to 3pi of sin^2(x) ?
The integral from 0 to 3pi of sin^2(x) is (3pi)/2