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

The integral from-1 to 0 of cos(npix) is 0

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integral from-1 to 0 of cos(npix)

\int_{- 1}^{0} cos (nπ x) d x

0

Solution steps

\int_{- 1}^{0} cos (nπ x) d x

Apply u-substitution

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

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

Take the constant out:

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

= \frac{1}{πn} \cdot \int_{- πn}^{0} cos (u) d u

Use the common integral:

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

= \frac{1}{πn} [sin (u)]_{- πn}^{0}

Compute the boundaries:

0

= \frac{1}{πn} \cdot 0

Simplify

= 0

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What is the integral from-1 to 0 of cos(npix) ?
The integral from-1 to 0 of cos(npix) is 0