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

Lösung

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

\frac{4}{3}

Dezimale

1.33333 \dots

Schritte zur Lösung

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

Vereinfache

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

Wende U-Substitution an

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

Wende U-Substitution an

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

Wende die Potenzregel an

= \frac{1}{2} [\frac{v ^{3}}{3}]_{0}^{2}

Berechne die Grenzen:

\frac{8}{3}

= \frac{1}{2} \cdot \frac{8}{3}

Vereinfache

= \frac{4}{3}

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