What is the integral of sin((npix)/3) ?

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

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integral of sin((npix)/3)

\int sin (\frac{nπ x}{3}) d x

- \frac{3}{πn} cos (\frac{πn x}{3}) + C

Solution steps

\int sin (\frac{nπ x}{3}) d x

Apply u-substitution

: \int \frac{3 sin ( u )}{πn} d u

= \int \frac{3 sin ( u )}{πn} d u

Take the constant out:

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

= \frac{3}{πn} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{3}{πn} (- cos (u))

Substitute back

u = \frac{n x π}{3}

= \frac{3}{πn} (- cos (\frac{n x π}{3}))

Simplify

= - \frac{3}{πn} cos (\frac{πn x}{3})

Add a constant to the solution

= - \frac{3}{πn} cos (\frac{πn x}{3}) + C

n = 0 \sum \infty \frac{1}{n + 3}

\int \frac{1}{2 x ^{5}} d x

x \to 0 lim (4 - x^{2})

\frac{x + x \cdot cos ( x ) - 2 \cdot sin ( x )}{x ^{3}}

\frac{d}{d x} (2 x^{4} + 2 y^{4} + z^{4} - 2 x^{2} y^{2} z)

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