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

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{x π}{L}

= \frac{L}{π} (- cos (\frac{x π}{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

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