What is the integral of-cos(npix) ?

The integral of-cos(npix) is -1/(pin)sin(npix)+C

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integral of-cos(npix)

\int - cos (nπ x) d x

- \frac{1}{πn} sin (nπ x) + C

Solution steps

\int - cos (nπ x) d x

Take the constant out:

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

= - \int cos (nπ x) d x

Apply u-substitution

: \int \frac{cos ( u )}{πn} d u

= - \int \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 cos (u) d u

Use the common integral:

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

= - \frac{1}{πn} sin (u)

Substitute back

u = nπ x

= - \frac{1}{πn} sin (nπ x)

Add a constant to the solution

= - \frac{1}{πn} sin (nπ x) + C

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What is the integral of-cos(npix) ?
The integral of-cos(npix) is -1/(pin)sin(npix)+C