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Beliebt Rechnen >

integral of (x^2+8x-3)/(x^3+3x^2)

Lösung

\int \frac{x ^{2} + 8 x - 3}{x ^{3} + 3 x ^{2}} d x

Lösung

3 ln ∣ x ∣ + \frac{1}{x} - 2 ln ∣ x + 3 ∣ + C

Schritte zur Lösung

\int \frac{x ^{2} + 8 x - 3}{x ^{3} + 3 x ^{2}} d x

Ermittle den Partialbruch von

\frac{x ^{2} + 8 x - 3}{x ^{3} + 3 x ^{2}} : \frac{3}{x} - \frac{1}{x ^{2}} - \frac{2}{x + 3}

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

\int \frac{2}{x + 3} d x = 2 ln ∣ x + 3 ∣

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

Vereinfache

= 3 ln ∣ x ∣ + \frac{1}{x} - 2 ln ∣ x + 3 ∣

Füge eine Konstante zur Lösung hinzu

= 3 ln ∣ x ∣ + \frac{1}{x} - 2 ln ∣ x + 3 ∣ + C

Graph

Beliebte Beispiele

integral of 1/(100-t)

\int \frac{1}{100 - t} d t

integral of 1/(x^2-8x+16)

\int \frac{1}{x ^{2} - 8 x + 16} d x

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\frac{d}{d x} (\frac{( x + 1 ) ^{2}}{x})

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\frac{\partial}{\partial z} (x \cdot sin (y) \cdot sin (z))