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Beliebt Rechnen >

integral of 1/((x-3)(x^2-2x)^2)

Lösung

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

Lösung

\frac{1}{9} ln ∣ x - 3 ∣ - \frac{1}{9} ln ∣ x ∣ + \frac{1}{12 x} + \frac{1}{4 ( x - 2 )} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{( x - 3 ) ( x ^{2} - 2 x ) ^{2}} : \frac{1}{9 ( x - 3 )} - \frac{1}{9 x} - \frac{1}{12 x ^{2}} - \frac{1}{4 ( x - 2 ) ^{2}}

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

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

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

\int \frac{1}{12 x ^{2}} d x = - \frac{1}{12 x}

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

= \frac{1}{9} ln ∣ x - 3 ∣ - \frac{1}{9} ln ∣ x ∣ - (- \frac{1}{12 x}) - (- \frac{1}{4 ( x - 2 )})

Vereinfache

= \frac{1}{9} ln ∣ x - 3 ∣ - \frac{1}{9} ln ∣ x ∣ + \frac{1}{12 x} + \frac{1}{4 ( x - 2 )}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{9} ln ∣ x - 3 ∣ - \frac{1}{9} ln ∣ x ∣ + \frac{1}{12 x} + \frac{1}{4 ( x - 2 )} + C

Graph

Beliebte Beispiele

integral of (3x+9)/(4x^2+x^3)

\int \frac{3 x + 9}{4 x ^{2} + x ^{3}} d x

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\int y^{\frac{- 1}{2}} d y

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\int (x^{\frac{3}{2}} + 2)^{7} x d x

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