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

The integral from 0 to pi/2 of (sin^2(θ))/2 is pi/8

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integral from 0 to pi/2 of (sin^2(θ))/2

\int_{0}^{\frac{π}{2}} \frac{sin ^{2} ( θ )}{2} d θ

\frac{π}{8}

Decimal

0.39269 \dots

Solution steps

\int_{0}^{\frac{π}{2}} \frac{sin ^{2} ( θ )}{2} d θ

Take the constant out:

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

= \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} sin^{2} (θ) d θ

Rewrite using trig identities

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

Take the constant out:

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

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

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} \cdot \frac{1}{2} (\int_{0}^{\frac{π}{2}} 1 d θ - \int_{0}^{\frac{π}{2}} cos (2 θ) d θ)

\int_{0}^{\frac{π}{2}} 1 d θ = \frac{π}{2}

\int_{0}^{\frac{π}{2}} cos (2 θ) d θ = 0

= \frac{1}{2} \cdot \frac{1}{2} (\frac{π}{2} - 0)

Simplify

\frac{1}{2} \cdot \frac{1}{2} (\frac{π}{2} - 0) : \frac{π}{8}

= \frac{π}{8}

What is the integral from 0 to pi/2 of (sin^2(θ))/2 ?
The integral from 0 to pi/2 of (sin^2(θ))/2 is pi/8