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Beliebt Rechnen >

integral of (x+5)/(2x^3-8x)

Lösung

\int \frac{x + 5}{2 x ^{3} - 8 x} d x

Lösung

\frac{1}{2} (- \frac{5}{4} ln ∣ x ∣ + \frac{3}{8} ln ∣ x + 2 ∣ + \frac{7}{8} ln ∣ x - 2 ∣) + C

Schritte zur Lösung

\int \frac{x + 5}{2 x ^{3} - 8 x} d x

Faktorisiere

\frac{x + 5}{2 x ^{3} - 8 x}

= \int \frac{x + 5}{2 x ( x + 2 ) ( x - 2 )} 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 + 5}{x ( x + 2 ) ( x - 2 )} d x

Ermittle den Partialbruch von

\frac{x + 5}{x ( x + 2 ) ( x - 2 )} : - \frac{5}{4 x} + \frac{3}{8 ( x + 2 )} + \frac{7}{8 ( x - 2 )}

= \frac{1}{2} \cdot \int - \frac{5}{4 x} + \frac{3}{8 ( x + 2 )} + \frac{7}{8 ( x - 2 )} 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

= \frac{1}{2} (- \int \frac{5}{4 x} d x + \int \frac{3}{8 ( x + 2 )} d x + \int \frac{7}{8 ( x - 2 )} d x)

\int \frac{5}{4 x} d x = \frac{5}{4} ln ∣ x ∣

\int \frac{3}{8 ( x + 2 )} d x = \frac{3}{8} ln ∣ x + 2 ∣

\int \frac{7}{8 ( x - 2 )} d x = \frac{7}{8} ln ∣ x - 2 ∣

= \frac{1}{2} (- \frac{5}{4} ln ∣ x ∣ + \frac{3}{8} ln ∣ x + 2 ∣ + \frac{7}{8} ln ∣ x - 2 ∣)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (- \frac{5}{4} ln ∣ x ∣ + \frac{3}{8} ln ∣ x + 2 ∣ + \frac{7}{8} ln ∣ x - 2 ∣) + C

Graph

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