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

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

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

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

\frac{8}{π}

Decimal

2.54647 \dots

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int sin (u) d u = - cos (u)

= 2 \cdot \frac{2}{π} [- cos (u)]_{0}^{π}

Simplify

2 \cdot \frac{2}{π} [- cos (u)]_{0}^{π} : \frac{4}{π} [- cos (u)]_{0}^{π}

= \frac{4}{π} [- cos (u)]_{0}^{π}

Compute the boundaries:

2

= \frac{4}{π} \cdot 2

Simplify

= \frac{8}{π}

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