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

The integral of sin((pix)/l) is -l/pi cos((pix)/l)+C

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integral of sin((pix)/l)

\int sin (\frac{π x}{l}) d x

- \frac{l}{π} cos (\frac{π x}{l}) + C

Solution steps

\int sin (\frac{π x}{l}) d x

Apply u-substitution

: \int \frac{l sin ( u )}{π} d u

= \int \frac{l sin ( u )}{π} d u

Take the constant out:

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

= \frac{l}{π} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{l}{π} (- cos (u))

Substitute back

u = \frac{π x}{l}

= \frac{l}{π} (- cos (\frac{π x}{l}))

Simplify

= - \frac{l}{π} cos (\frac{π x}{l})

Add a constant to the solution

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

\int cos (\frac{2 π x}{L}) d x

\int \frac{2 ( 1 + x )}{( 1 - 2 x - x ^{2} )} d x

\int 2 x^{2} d x

\int ln (4) x^{2} d x

\int \frac{( e ^{u} )}{( 4 - e ^{u} ) ^{2}} d u

What is the integral of sin((pix)/l) ?
The integral of sin((pix)/l) is -l/pi cos((pix)/l)+C