integral of ((ln(x)-2)^3(ln(x)+1))/x

Lösung

\int \frac{( ln ( x ) - 2 ) ^{3} ( ln ( x ) + 1 )}{x} d x

\frac{ln ^{5} ( x )}{5} - \frac{5}{4} ln^{4} (x) + 2 ln^{3} (x) + 2 ln^{2} (x) - 8 ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{( ln ( x ) - 2 ) ^{3} ( ln ( x ) + 1 )}{x} d x

Multipliziere aus

\frac{( ln ( x ) - 2 ) ^{3} ( ln ( x ) + 1 )}{x} : \frac{ln ^{4} ( x )}{x} - \frac{5 ln ^{3} ( x )}{x} + \frac{6 ln ^{2} ( x )}{x} + \frac{4 ln ( x )}{x} - \frac{8}{x}

= \int \frac{ln ^{4} ( x )}{x} - \frac{5 ln ^{3} ( x )}{x} + \frac{6 ln ^{2} ( x )}{x} + \frac{4 ln ( x )}{x} - \frac{8}{x} d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{ln ^{4} ( x )}{x} d x - \int \frac{5 ln ^{3} ( x )}{x} d x + \int \frac{6 ln ^{2} ( x )}{x} d x + \int \frac{4 ln ( x )}{x} d x - \int \frac{8}{x} d x

\int \frac{ln ^{4} ( x )}{x} d x = \frac{ln ^{5} ( x )}{5}

\int \frac{5 ln ^{3} ( x )}{x} d x = \frac{5}{4} ln^{4} (x)

\int \frac{6 ln ^{2} ( x )}{x} d x = 2 ln^{3} (x)

\int \frac{4 ln ( x )}{x} d x = 2 ln^{2} (x)

\int \frac{8}{x} d x = 8 ln ∣ x ∣

= \frac{ln ^{5} ( x )}{5} - \frac{5}{4} ln^{4} (x) + 2 ln^{3} (x) + 2 ln^{2} (x) - 8 ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= \frac{ln ^{5} ( x )}{5} - \frac{5}{4} ln^{4} (x) + 2 ln^{3} (x) + 2 ln^{2} (x) - 8 ln ∣ x ∣ + C

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