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Beliebt Rechnen >

integral of (81)/(x^4+27x)

Lösung

\int \frac{81}{x ^{4} + 27 x} d x

Lösung

3 ln ∣ x ∣ - ln ∣ x + 3 ∣ - ln 4 x^{2} - 12 x + 36 + C

Schritte zur Lösung

\int \frac{81}{x ^{4} + 27 x} d x

Entferne die Konstante:

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

= 81 \cdot \int \frac{1}{x ^{4} + 27 x} d x

Ermittle den Partialbruch von

\frac{1}{x ^{4} + 27 x} : \frac{1}{27 x} - \frac{1}{81 ( x + 3 )} + \frac{- 2 x + 3}{81 ( x ^{2} - 3 x + 9 )}

= 81 \cdot \int \frac{1}{27 x} - \frac{1}{81 ( x + 3 )} + \frac{- 2 x + 3}{81 ( x ^{2} - 3 x + 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

= 81 (\int \frac{1}{27 x} d x - \int \frac{1}{81 ( x + 3 )} d x + \int \frac{- 2 x + 3}{81 ( x ^{2} - 3 x + 9 )} d x)

\int \frac{1}{27 x} d x = \frac{1}{27} ln ∣ x ∣

\int \frac{1}{81 ( x + 3 )} d x = \frac{1}{81} ln ∣ x + 3 ∣

\int \frac{- 2 x + 3}{81 ( x ^{2} - 3 x + 9 )} d x = - \frac{1}{81} ln 4 x^{2} - 12 x + 36

= 81 (\frac{1}{27} ln ∣ x ∣ - \frac{1}{81} ln ∣ x + 3 ∣ - \frac{1}{81} ln 4 x^{2} - 12 x + 36)

Vereinfache

81 (\frac{1}{27} ln ∣ x ∣ - \frac{1}{81} ln ∣ x + 3 ∣ - \frac{1}{81} ln 4 x^{2} - 12 x + 36) : 3 ln ∣ x ∣ - ln ∣ x + 3 ∣ - ln 4 x^{2} - 12 x + 36

= 3 ln ∣ x ∣ - ln ∣ x + 3 ∣ - ln 4 x^{2} - 12 x + 36

Füge eine Konstante zur Lösung hinzu

= 3 ln ∣ x ∣ - ln ∣ x + 3 ∣ - ln 4 x^{2} - 12 x + 36 + C

Graph

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