What is the integral from 0 to l of wcos(pi/(2l)x) ?

The integral from 0 to l of wcos(pi/(2l)x) is (2lw)/pi

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integral from 0 to l of wcos(pi/(2l)x)

\int_{0}^{l} w cos (\frac{π}{2 l} x) d x

\frac{2 lw}{π}

Solution steps

\int_{0}^{l} w cos (\frac{π}{2 l} x) d x

Take the constant out:

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

= w \cdot \int_{0}^{l} cos (\frac{π}{2 l} x) d x

\frac{π}{2 l} = π \frac{1}{2 l}

= w \cdot \int_{0}^{l} cos (π \frac{1}{2 l} x) d x

Multiply

π \frac{1}{2 l} x : \frac{π x}{2 l}

= w \cdot \int_{0}^{l} cos (\frac{π x}{2 l}) d x

Apply u-substitution

= w \cdot \int_{0}^{\frac{π}{2}} \frac{2 l cos ( u )}{π} d u

Take the constant out:

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

= w \frac{2 l}{π} \cdot \int_{0}^{\frac{π}{2}} cos (u) d u

Use the common integral:

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

= w \frac{2 l}{π} [sin (u)]_{0}^{\frac{π}{2}}

Compute the boundaries:

1

= w \frac{2 l}{π} \cdot 1

Simplify

= \frac{2 lw}{π}

What is the integral from 0 to l of wcos(pi/(2l)x) ?
The integral from 0 to l of wcos(pi/(2l)x) is (2lw)/pi