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

Lösung

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

\frac{2 lw}{π}

Schritte zur Lösung

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

Entferne die Konstante:

\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

Multipliziere

π \frac{1}{2 l} x : \frac{π x}{2 l}

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

Wende U-Substitution an

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

Entferne die Konstante:

\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

Nutze das gemeinsame Integral :

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

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

Berechne die Grenzen:

1

= w \frac{2 l}{π} \cdot 1

Vereinfache

= \frac{2 lw}{π}

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