What is the integral of cos((2npix)/3) ?

The integral of cos((2npix)/3) is 3/(2pin)sin((2pinx)/3)+C

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integral of cos((2npix)/3)

\int cos (\frac{2 nπ x}{3}) d x

\frac{3}{2 πn} sin (\frac{2 πn x}{3}) + C

Solution steps

\int cos (\frac{2 nπ x}{3}) d x

Apply u-substitution

: \int \frac{cos ( \frac{u}{3} )}{2 πn} d u

= \int \frac{cos ( \frac{u}{3} )}{2 πn} d u

Take the constant out:

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

= \frac{1}{2 πn} \cdot \int cos (\frac{u}{3}) d u

Apply u-substitution

: \int cos (v) \cdot 3 d v

= \frac{1}{2 πn} \cdot \int cos (v) \cdot 3 d v

Take the constant out:

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

= \frac{1}{2 πn} \cdot 3 \cdot \int cos (v) d v

Use the common integral:

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

= \frac{1}{2 πn} \cdot 3 sin (v)

Substitute back

= \frac{1}{2 πn} \cdot 3 sin (\frac{2 n x π}{3})

Simplify

\frac{1}{2 πn} \cdot 3 sin (\frac{2 n x π}{3}) : \frac{3}{2 πn} sin (\frac{2 πn x}{3})

= \frac{3}{2 πn} sin (\frac{2 πn x}{3})

Add a constant to the solution

= \frac{3}{2 πn} sin (\frac{2 πn x}{3}) + C

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What is the integral of cos((2npix)/3) ?
The integral of cos((2npix)/3) is 3/(2pin)sin((2pinx)/3)+C