What is the integral of sin((pix)/(2y)) ?

The integral of sin((pix)/(2y)) is -(2y)/pi cos((pix)/(2y))+C

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integral of sin((pix)/(2y))

\int sin (\frac{π x}{2 y}) d x

- \frac{2 y}{π} cos (\frac{π x}{2 y}) + C

Solution steps

\int sin (\frac{π x}{2 y}) d x

Apply u-substitution

: \int \frac{2 y sin ( u )}{π} d u

= \int \frac{2 y sin ( u )}{π} d u

Take the constant out:

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

= \frac{2 y}{π} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{2 y}{π} (- cos (u))

Substitute back

u = \frac{x π}{2 y}

= \frac{2 y}{π} (- cos (\frac{x π}{2 y}))

Simplify

= - \frac{2 y}{π} cos (\frac{π x}{2 y})

Add a constant to the solution

= - \frac{2 y}{π} cos (\frac{π x}{2 y}) + C

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\int \frac{5 x ^{2} - 8 x - 8}{5 x} d x

\int cosh (\frac{x}{3}) d x

What is the integral of sin((pix)/(2y)) ?
The integral of sin((pix)/(2y)) is -(2y)/pi cos((pix)/(2y))+C