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Beliebt Rechnen >

integral of (18)/((x-3)(x^2+9))

Lösung

\int \frac{18}{( x - 3 ) ( x ^{2} + 9 )} d x

Lösung

ln ∣ x - 3 ∣ - \frac{1}{2} ln x^{2} + 9 - arctan (\frac{x}{3}) + C

Schritte zur Lösung

\int \frac{18}{( x - 3 ) ( x ^{2} + 9 )} d x

Entferne die Konstante:

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

= 18 \cdot \int \frac{1}{( x - 3 ) ( x ^{2} + 9 )} d x

Ermittle den Partialbruch von

\frac{1}{( x - 3 ) ( x ^{2} + 9 )} : \frac{1}{18 ( x - 3 )} + \frac{- x - 3}{18 ( x ^{2} + 9 )}

= 18 \cdot \int \frac{1}{18 ( x - 3 )} + \frac{- x - 3}{18 ( x ^{2} + 9 )} 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

= 18 (\int \frac{1}{18 ( x - 3 )} d x + \int \frac{- x - 3}{18 ( x ^{2} + 9 )} d x)

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

\int \frac{- x - 3}{18 ( x ^{2} + 9 )} d x = \frac{1}{18} (- \frac{1}{2} ln x^{2} + 9 - arctan (\frac{x}{3}))

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

Vereinfache

18 (\frac{1}{18} ln ∣ x - 3 ∣ + \frac{1}{18} (- \frac{1}{2} ln x^{2} + 9 - arctan (\frac{x}{3}))) : ln ∣ x - 3 ∣ - \frac{1}{2} ln x^{2} + 9 - arctan (\frac{x}{3})

= ln ∣ x - 3 ∣ - \frac{1}{2} ln x^{2} + 9 - arctan (\frac{x}{3})

Füge eine Konstante zur Lösung hinzu

= ln ∣ x - 3 ∣ - \frac{1}{2} ln x^{2} + 9 - arctan (\frac{x}{3}) + C

Graph

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