What is the integral of pi/(4W)cos((pix)/(2W)) ?

The integral of pi/(4W)cos((pix)/(2W)) is 1/2 sin((pix)/(2W))+C

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integral of pi/(4W)cos((pix)/(2W))

\int \frac{π}{4 W} cos (\frac{π x}{2 W}) d x

\frac{1}{2} sin (\frac{π x}{2 W}) + C

Solution steps

\int \frac{π}{4 W} cos (\frac{π x}{2 W}) d x

Take the constant out:

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

= \frac{π}{4 W} \cdot \int cos (\frac{x π}{2 W}) d x

Apply u-substitution

= \frac{π}{4 W} \cdot \int \frac{2 W cos ( u )}{π} d u

Take the constant out:

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

= \frac{π}{4 W} \cdot \frac{2 W}{π} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{π}{4 W} \cdot \frac{2 W}{π} sin (u)

Substitute back

u = \frac{x π}{2 W}

= \frac{π}{4 W} \cdot \frac{2 W}{π} sin (\frac{x π}{2 W})

Simplify

\frac{π}{4 W} \cdot \frac{2 W}{π} sin (\frac{x π}{2 W}) : \frac{1}{2} sin (\frac{π x}{2 W})

= \frac{1}{2} sin (\frac{π x}{2 W})

Add a constant to the solution

= \frac{1}{2} sin (\frac{π x}{2 W}) + C

What is the integral of pi/(4W)cos((pix)/(2W)) ?
The integral of pi/(4W)cos((pix)/(2W)) is 1/2 sin((pix)/(2W))+C