integral from 1 to 2 of ycos(xy)

Lösung

\int_{1}^{2} y cos (x y) d x

sin (2 y) - sin (y)

Schritte zur Lösung

\int_{1}^{2} y cos (x y) d x

Entferne die Konstante:

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

= y \cdot \int_{1}^{2} cos (y x) d x

Wende U-Substitution an

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

Entferne die Konstante:

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

= y \frac{1}{y} \cdot \int_{y}^{y \cdot 2} cos (u) d u

Nutze das gemeinsame Integral :

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

= y \frac{1}{y} [sin (u)]_{y}^{y \cdot 2}

Vereinfache

y \frac{1}{y} [sin (u)]_{y}^{y \cdot 2} : [sin (u)]_{y}^{2 y}

= [sin (u)]_{y}^{2 y}

Berechne die Grenzen:

sin (2 y) - sin (y)

= sin (2 y) - sin (y)

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