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

The integral from 0 to 2 of (sin((pix)/2))^2 is 1

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integral from 0 to 2 of (sin((pix)/2))^2

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

1

Solution steps

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

Simplify

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

Rewrite using trig identities

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

\int_{0}^{2} 1 d x = 2

\int_{0}^{2} cos (π x) d x = 0

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

Simplify

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

= 1

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\int_{0}^{2} (\frac{x}{2})^{2} d x

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What is the integral from 0 to 2 of (sin((pix)/2))^2 ?
The integral from 0 to 2 of (sin((pix)/2))^2 is 1