What is the integral of cos((npix)/pi) ?

The integral of cos((npix)/pi) is 1/n sin(nx)+C

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integral of cos((npix)/pi)

\int cos (\frac{nπ x}{π}) d x

\frac{1}{n} sin (n x) + C

Solution steps

\int cos (\frac{nπ x}{π}) d x

Cancel the common factor:

π

= \int cos (n x) d x

Apply u-substitution

= \int \frac{cos ( u )}{n} d u

Take the constant out:

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

= \frac{1}{n} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{n} sin (u)

Substitute back

u = n x

= \frac{1}{n} sin (n x)

Add a constant to the solution

= \frac{1}{n} sin (n x) + C

\int \frac{3 x}{x - 1} d x

\int 4 cos (\frac{1}{4} π x) d x

\int \frac{- 1}{2} d x

\int \frac{x + 1}{2 x ^{2} + 2 x} d x

\int \frac{1}{8 x - 2} d x

What is the integral of cos((npix)/pi) ?
The integral of cos((npix)/pi) is 1/n sin(nx)+C