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Beliebt Rechnen >

integral of (x^2+30x-9)/(x^3-9x)

Lösung

\int \frac{x ^{2} + 30 x - 9}{x ^{3} - 9 x} d x

Lösung

ln ∣ x ∣ - 5 ln ∣ x + 3 ∣ + 5 ln ∣ x - 3 ∣ + C

Schritte zur Lösung

\int \frac{x ^{2} + 30 x - 9}{x ^{3} - 9 x} d x

Ermittle den Partialbruch von

\frac{x ^{2} + 30 x - 9}{x ^{3} - 9 x} : \frac{1}{x} - \frac{5}{x + 3} + \frac{5}{x - 3}

= \int \frac{1}{x} - \frac{5}{x + 3} + \frac{5}{x - 3} 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{1}{x} d x - \int \frac{5}{x + 3} d x + \int \frac{5}{x - 3} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{5}{x + 3} d x = 5 ln ∣ x + 3 ∣

\int \frac{5}{x - 3} d x = 5 ln ∣ x - 3 ∣

= ln ∣ x ∣ - 5 ln ∣ x + 3 ∣ + 5 ln ∣ x - 3 ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ - 5 ln ∣ x + 3 ∣ + 5 ln ∣ x - 3 ∣ + C

Graph

Beliebte Beispiele

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\int \frac{19}{( x ^{2} - 1 ) ^{2}} d x

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\int \frac{arcsin ( x )}{2 x ^{2}} d x

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\int t^{9} d t