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

Lösung

\int_{y}^{2} sin (\frac{π x}{2 y}) d x

- \frac{2 y cos ( \frac{π}{y} )}{π}

Schritte zur Lösung

\int_{y}^{2} sin (\frac{π x}{2 y}) d x

Wende U-Substitution an

= \int_{\frac{π}{2}}^{\frac{π}{y}} \frac{2 y sin ( u )}{π} d u

Entferne die Konstante:

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

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

Nutze das gemeinsame Integral :

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

= \frac{2 y}{π} [- cos (u)]_{\frac{π}{2}}^{\frac{π}{y}}

Berechne die Grenzen:

- cos (\frac{π}{y})

= \frac{2 y}{π} (- cos (\frac{π}{y}))

Vereinfache

= - \frac{2 y cos ( \frac{π}{y} )}{π}

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