integral from 1 to e of (ln(x^2))/(2x)

Lösung

\int_{1}^{e} \frac{ln ( x ^{2} )}{2 x} d x

\frac{1}{2}

Dezimale

0.5

Schritte zur Lösung

\int_{1}^{e} \frac{ln ( x ^{2} )}{2 x} d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{1}^{e} \frac{ln ( x ^{2} )}{x} d x

Wende U-Substitution an

= \frac{1}{2} \cdot \int_{1}^{e^{2}} \frac{ln ( u )}{2 u} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int_{1}^{e^{2}} \frac{ln ( u )}{u} d u

Wende U-Substitution an

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int_{0}^{2} v d v

Wende die Potenzregel an

= \frac{1}{2} \cdot \frac{1}{2} [\frac{v ^{2}}{2}]_{0}^{2}

Vereinfache

\frac{1}{2} \cdot \frac{1}{2} [\frac{v ^{2}}{2}]_{0}^{2} : \frac{1}{4} [\frac{v ^{2}}{2}]_{0}^{2}

= \frac{1}{4} [\frac{v ^{2}}{2}]_{0}^{2}

Berechne die Grenzen:

2

= \frac{1}{4} \cdot 2

Vereinfache

= \frac{1}{2}

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