What is the integral of cos((npix)/(2pi)) ?

The integral of cos((npix)/(2pi)) is 2/n sin((nx)/2)+C

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integral of cos((npix)/(2pi))

\int cos (\frac{nπ x}{2 π}) d x

\frac{2}{n} sin (\frac{n x}{2}) + C

Solution steps

\int cos (\frac{nπ x}{2 π}) d x

Cancel the common factor:

π

= \int cos (\frac{n x}{2}) d x

Apply u-substitution

: \int \frac{2 cos ( u )}{n} d u

= \int \frac{2 cos ( u )}{n} d u

Take the constant out:

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

= \frac{2}{n} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{2}{n} sin (u)

Substitute back

u = \frac{n x}{2}

= \frac{2}{n} sin (\frac{n x}{2})

Add a constant to the solution

= \frac{2}{n} sin (\frac{n x}{2}) + C

x \to 2 - lim (\frac{3}{2 - x})

f^{^{'}} (x) = 2 x^{2} + x - 1

\int 64 - x^{2} d x

\frac{d}{d x} (\frac{5}{x + 8})

\frac{d}{d x} (ln (4 sin^{2} (x)))

What is the integral of cos((npix)/(2pi)) ?
The integral of cos((npix)/(2pi)) is 2/n sin((nx)/2)+C