integral from e^x to e^{2x of}xln(y)

Lösung

\int_{e^{x}}^{e^{2 x}} x ln (y) d y

x (2 x e^{2 x} - e^{2 x} - x e^{x} + e^{x})

Schritte zur Lösung

\int_{e^{x}}^{e^{2 x}} x ln (y) d y

Entferne die Konstante:

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

= x \cdot \int_{e^{x}}^{e^{2 x}} ln (y) d y

Wende die partielle Integration an

= x [y ln (y) - \int 1 d y]_{e^{x}}^{e^{2 x}}

\int 1 d y = y

= x [y ln (y) - y]_{e^{x}}^{e^{2 x}}

Berechne die Grenzen:

2 x e^{2 x} - e^{2 x} - x e^{x} + e^{x}

= x (2 x e^{2 x} - e^{2 x} - x e^{x} + e^{x})

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