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Beliebt Rechnen >

integral of 1/(x^3+6x^2+9x)

Lösung

\int \frac{1}{x ^{3} + 6 x ^{2} + 9 x} d x

Lösung

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

Schritte zur Lösung

\int \frac{1}{x ^{3} + 6 x ^{2} + 9 x} d x

Ermittle den Partialbruch von

\frac{1}{x ^{3} + 6 x ^{2} + 9 x} : \frac{1}{9 x} - \frac{1}{9 ( x + 3 )} - \frac{1}{3 ( x + 3 ) ^{2}}

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

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

\int \frac{1}{9 x} d x = \frac{1}{9} ln ∣ x ∣

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of cos(x)cos(3x)cos(5x)

\int cos (x) cos (3 x) cos (5 x) d x

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\int \frac{1}{36 + ( x - 4 ) ^{2}} d x

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\int 3 x^{2} (x^{3} - 3)^{4} d x

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\int 8 x ln (6 x) d x

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\int \frac{x ^{3} + x ^{2}}{x} d x