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Beliebt Rechnen >

integral of (x+4)/(2x^3-18x)

Lösung

\int \frac{x + 4}{2 x ^{3} - 18 x} d x

Lösung

\frac{1}{36} (ln ∣ x + 3 ∣ - 8 ln ∣ x ∣ + 7 ln ∣ x - 3 ∣) + C

Schritte zur Lösung

\int \frac{x + 4}{2 x ^{3} - 18 x} d x

Faktorisiere

\frac{x + 4}{2 x ^{3} - 18 x}

= \int \frac{x + 4}{2 x ( x + 3 ) ( x - 3 )} d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{x + 4}{x ( x + 3 ) ( x - 3 )} d x

Multipliziere aus

\frac{x + 4}{x ( x + 3 ) ( x - 3 )} : \frac{x}{x ( x + 3 ) ( x - 3 )} + \frac{4}{x ( x + 3 ) ( x - 3 )}

Wende die Summenregel an:

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

= \frac{1}{2} (\int \frac{x}{x ( x + 3 ) ( x - 3 )} d x + \int \frac{4}{x ( x + 3 ) ( x - 3 )} d x)

\int \frac{x}{x ( x + 3 ) ( x - 3 )} d x = - \frac{1}{6} ln ∣ x + 3 ∣ + \frac{1}{6} ln ∣ x - 3 ∣

\int \frac{4}{x ( x + 3 ) ( x - 3 )} d x = 4 (\frac{1}{18} ln ∣ x + 3 ∣ - \frac{1}{9} ln ∣ x ∣ + \frac{1}{18} ln ∣ x - 3 ∣)

= \frac{1}{2} (- \frac{1}{6} ln ∣ x + 3 ∣ + \frac{1}{6} ln ∣ x - 3 ∣ + 4 (\frac{1}{18} ln ∣ x + 3 ∣ - \frac{1}{9} ln ∣ x ∣ + \frac{1}{18} ln ∣ x - 3 ∣))

Vereinfache

\frac{1}{2} (- \frac{1}{6} ln ∣ x + 3 ∣ + \frac{1}{6} ln ∣ x - 3 ∣ + 4 (\frac{1}{18} ln ∣ x + 3 ∣ - \frac{1}{9} ln ∣ x ∣ + \frac{1}{18} ln ∣ x - 3 ∣)) : \frac{1}{36} (ln ∣ x + 3 ∣ - 8 ln ∣ x ∣ + 7 ln ∣ x - 3 ∣)

= \frac{1}{36} (ln ∣ x + 3 ∣ - 8 ln ∣ x ∣ + 7 ln ∣ x - 3 ∣)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{36} (ln ∣ x + 3 ∣ - 8 ln ∣ x ∣ + 7 ln ∣ x - 3 ∣) + C

Graph

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