integral from 0 to ln(a) of 4e^{2x}

Lösung

\int_{0}^{l n (a)} 4 e^{2 x} d x

2 (a^{2} - 1)

Schritte zur Lösung

\int_{0}^{l n (a)} 4 e^{2 x} d x

Entferne die Konstante:

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

= 4 \cdot \int_{0}^{l n (a)} e^{2 x} d x

Wende U-Substitution an

= 4 \cdot \int_{0}^{2 l n (a)} e^{u} \frac{1}{2} d u

Entferne die Konstante:

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

= 4 \cdot \frac{1}{2} \cdot \int_{0}^{2 l n (a)} e^{u} d u

Nutze das gemeinsame Integral :

\int e^{u} d u = e^{u}

= 4 \cdot \frac{1}{2} [e^{u}]_{0}^{2 l n (a)}

Vereinfache

4 \cdot \frac{1}{2} [e^{u}]_{0}^{2 l n (a)} : 2 [e^{u}]_{0}^{2 l n (a)}

= 2 [e^{u}]_{0}^{2 l n (a)}

Berechne die Grenzen:

a^{2} - 1

= 2 (a^{2} - 1)

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