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Beliebt Rechnen >

integral from 0 to 1 of (ln(x))/(5x)

Lösung

\int_{0}^{1} \frac{ln ( x )}{5 x} d x

Lösung

divergiert

Schritte zur Lösung

\int_{0}^{1} \frac{ln ( x )}{5 x} d x

Entferne die Konstante:

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

= \frac{1}{5} \cdot \int_{0}^{1} \frac{ln ( x )}{x} d x

Wende die partielle Integration an

= \frac{1}{5} [ln (x) - \int \frac{1 - ln ( x )}{x} d x]_{0}^{1}

\int \frac{1 - ln ( x )}{x} d x = - \frac{( 1 - ln ( x ) ) ^{2}}{2}

= \frac{1}{5} [ln (x) - (- \frac{( 1 - ln ( x ) ) ^{2}}{2})]_{0}^{1}

Vereinfache

\frac{1}{5} [ln (x) - (- \frac{( 1 - ln ( x ) ) ^{2}}{2})]_{0}^{1} : \frac{1}{5} [\frac{ln ^{2} ( x ) + 1}{2}]_{0}^{1}

= \frac{1}{5} [\frac{ln ^{2} ( x ) + 1}{2}]_{0}^{1}

Berechne die Grenzen:

divergiert

= divergiert

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