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Beliebt Rechnen >

integral of 4/(x(x-3)(x+1)^2)

Lösung

\int \frac{4}{x ( x - 3 ) ( x + 1 ) ^{2}} d x

Lösung

- \frac{4}{3} ln ∣ x ∣ + \frac{1}{12} ln ∣ x - 3 ∣ + \frac{5}{4} ln ∣ x + 1 ∣ - \frac{1}{x + 1} + C

Schritte zur Lösung

\int \frac{4}{x ( x - 3 ) ( x + 1 ) ^{2}} d x

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

\frac{1}{x ( x - 3 ) ( x + 1 ) ^{2}} : - \frac{1}{3 x} + \frac{1}{48 ( x - 3 )} + \frac{5}{16 ( x + 1 )} + \frac{1}{4 ( x + 1 ) ^{2}}

= 4 \cdot \int - \frac{1}{3 x} + \frac{1}{48 ( x - 3 )} + \frac{5}{16 ( x + 1 )} + \frac{1}{4 ( x + 1 ) ^{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

= 4 (- \int \frac{1}{3 x} d x + \int \frac{1}{48 ( x - 3 )} d x + \int \frac{5}{16 ( x + 1 )} d x + \int \frac{1}{4 ( x + 1 ) ^{2}} d x)

\int \frac{1}{3 x} d x = \frac{1}{3} ln ∣ x ∣

\int \frac{1}{48 ( x - 3 )} d x = \frac{1}{48} ln ∣ x - 3 ∣

\int \frac{5}{16 ( x + 1 )} d x = \frac{5}{16} ln ∣ x + 1 ∣

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

= 4 (- \frac{1}{3} ln ∣ x ∣ + \frac{1}{48} ln ∣ x - 3 ∣ + \frac{5}{16} ln ∣ x + 1 ∣ - \frac{1}{4 ( x + 1 )})

Vereinfache

4 (- \frac{1}{3} ln ∣ x ∣ + \frac{1}{48} ln ∣ x - 3 ∣ + \frac{5}{16} ln ∣ x + 1 ∣ - \frac{1}{4 ( x + 1 )}) : - \frac{4}{3} ln ∣ x ∣ + \frac{1}{12} ln ∣ x - 3 ∣ + \frac{5}{4} ln ∣ x + 1 ∣ - \frac{1}{x + 1}

= - \frac{4}{3} ln ∣ x ∣ + \frac{1}{12} ln ∣ x - 3 ∣ + \frac{5}{4} ln ∣ x + 1 ∣ - \frac{1}{x + 1}

Füge eine Konstante zur Lösung hinzu

= - \frac{4}{3} ln ∣ x ∣ + \frac{1}{12} ln ∣ x - 3 ∣ + \frac{5}{4} ln ∣ x + 1 ∣ - \frac{1}{x + 1} + C

Graph

Beliebte Beispiele

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

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

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\int \frac{tan ^{3} ( θ )}{sec ^{4} ( θ )} d θ

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\int x^{2} \cdot cos (n \cdot x) d x

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