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

The integral of 2sin((pix)/2) is -4/pi cos((pix)/2)+C

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integral of 2sin((pix)/2)

\int 2 sin (\frac{π x}{2}) d x

- \frac{4}{π} cos (\frac{π x}{2}) + C

Solution steps

\int 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 sin (\frac{π x}{2}) d x

Apply u-substitution

: \int sin (u) \frac{2}{π} d u

= 2 \cdot \int 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 sin (u) d u

Use the common integral:

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

= 2 \cdot \frac{2}{π} (- cos (u))

Substitute back

u = \frac{π x}{2}

= 2 \cdot \frac{2}{π} (- cos (\frac{π x}{2}))

Simplify

2 \cdot \frac{2}{π} (- cos (\frac{π x}{2})) : - \frac{4}{π} cos (\frac{π x}{2})

= - \frac{4}{π} cos (\frac{π x}{2})

Add a constant to the solution

= - \frac{4}{π} cos (\frac{π x}{2}) + C

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What is the integral of 2sin((pix)/2) ?
The integral of 2sin((pix)/2) is -4/pi cos((pix)/2)+C