integral from 1 to e^5 of (ln^3(x^2))/x

Lösung

\int_{1}^{e^{5}} \frac{ln ^{3} ( x ^{2} )}{x} d x

1250

Schritte zur Lösung

\int_{1}^{e^{5}} \frac{ln ^{3} ( x ^{2} )}{x} d x

Vereinfache

\frac{ln ^{3} ( x ^{2} )}{x} : \frac{8 ln ^{3} ( x )}{x}

= \int_{1}^{e^{5}} \frac{8 ln ^{3} ( x )}{x} d x

Entferne die Konstante:

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

= 8 \cdot \int_{1}^{e^{5}} \frac{ln ^{3} ( x )}{x} d x

Wende U-Substitution an

= 8 \cdot \int_{0}^{5} u^{3} d u

Wende die Potenzregel an

= 8 [\frac{u ^{4}}{4}]_{0}^{5}

Berechne die Grenzen:

\frac{625}{4}

= 8 \cdot \frac{625}{4}

Vereinfache

= 1250

\int x^{- \frac{1}{4}} 1 + x^{\frac{1}{4}} d x

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