What is the integral from-2 to 2 of cos(x-(npix)/2) ?

The integral from-2 to 2 of cos(x-(npix)/2) is (2(sin(2)(-1)^{-n}+sin(2)(-1)^n))/(2-pin)

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integral from-2 to 2 of cos(x-(npix)/2)

\int_{- 2}^{2} cos (x - \frac{nπ x}{2}) d x

\frac{2 ( sin ( 2 ) ( - 1 ) ^{- n} + sin ( 2 ) ( - 1 ) ^{n} )}{2 - πn}

Solution steps

\int_{- 2}^{2} cos (x - \frac{nπ x}{2}) d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Compute the boundaries:

sin (2) (- 1)^{- n} + sin (2) (- 1)^{n}

= \frac{2}{2 - πn} (sin (2) (- 1)^{- n} + sin (2) (- 1)^{n})

Simplify

= \frac{2 ( sin ( 2 ) ( - 1 ) ^{- n} + sin ( 2 ) ( - 1 ) ^{n} )}{2 - πn}

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What is the integral from-2 to 2 of cos(x-(npix)/2) ?
The integral from-2 to 2 of cos(x-(npix)/2) is (2(sin(2)(-1)^{-n}+sin(2)(-1)^n))/(2-pin)