What is the integral from 0 to 1 of cos((npix)/2) ?

The integral from 0 to 1 of cos((npix)/2) is (2sin((pin)/2))/(pin)

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integral from 0 to 1 of cos((npix)/2)

\int_{0}^{1} cos (\frac{nπ x}{2}) d x

\frac{2 sin ( \frac{πn}{2} )}{πn}

Solution steps

\int_{0}^{1} cos (\frac{nπ x}{2}) d x

Apply u-substitution

: \int_{0}^{\frac{πn}{2}} \frac{2 cos ( u )}{πn} d u

= \int_{0}^{\frac{πn}{2}} \frac{2 cos ( u )}{πn} d u

Take the constant out:

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

= \frac{2}{πn} \cdot \int_{0}^{\frac{πn}{2}} cos (u) d u

Use the common integral:

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

= \frac{2}{πn} [sin (u)]_{0}^{\frac{πn}{2}}

Compute the boundaries:

sin (\frac{πn}{2})

= \frac{2}{πn} sin (\frac{πn}{2})

Simplify

= \frac{2 sin ( \frac{πn}{2} )}{πn}

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What is the integral from 0 to 1 of cos((npix)/2) ?
The integral from 0 to 1 of cos((npix)/2) is (2sin((pin)/2))/(pin)