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

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

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integral of sin((npix)/l)

\int sin (\frac{nπ x}{l}) d x

- l \frac{1}{πn} cos (\frac{πn x}{l}) + C

Solution steps

\int sin (\frac{nπ x}{l}) d x

Apply u-substitution

: \int \frac{sin ( \frac{u}{l} )}{πn} d u

= \int \frac{sin ( \frac{u}{l} )}{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 sin (\frac{u}{l}) d u

Apply u-substitution

: \int sin (v) l d v

= \frac{1}{πn} \cdot \int sin (v) l d v

Take the constant out:

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

= \frac{1}{πn} l \cdot \int sin (v) d v

Use the common integral:

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

= \frac{1}{πn} l (- cos (v))

Substitute back

= \frac{1}{πn} l (- cos (\frac{nπ x}{l}))

Simplify

= - l \frac{1}{πn} cos (\frac{πn x}{l})

Add a constant to the solution

= - l \frac{1}{πn} cos (\frac{πn x}{l}) + C

\int_{0}^{3 y - 5} x y^{2} d x

\int e^{- y^{2}} d y

y = \frac{2}{x}, (- 3, - \frac{2}{3})

\frac{d}{d x} (- x^{3} + 3 x^{2} + 4)

y^{^{'}} y^{^{''}} = 2

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