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Beliebt Rechnen >

integral of (x^2+8x+4)/(x^4+2x^3+x^2)

Lösung

\int \frac{x ^{2} + 8 x + 4}{x ^{4} + 2 x ^{3} + x ^{2}} d x

Lösung

\frac{3}{x + 1} - \frac{4}{x} + C

Schritte zur Lösung

\int \frac{x ^{2} + 8 x + 4}{x ^{4} + 2 x ^{3} + x ^{2}} d x

Multipliziere aus

\frac{x ^{2} + 8 x + 4}{x ^{4} + 2 x ^{3} + x ^{2}} : \frac{x ^{2}}{x ^{4} + 2 x ^{3} + x ^{2}} + \frac{8 x}{x ^{4} + 2 x ^{3} + x ^{2}} + \frac{4}{x ^{4} + 2 x ^{3} + x ^{2}}

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{x ^{2}}{x ^{4} + 2 x ^{3} + x ^{2}} d x + \int \frac{8 x}{x ^{4} + 2 x ^{3} + x ^{2}} d x + \int \frac{4}{x ^{4} + 2 x ^{3} + x ^{2}} d x

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

\int \frac{8 x}{x ^{4} + 2 x ^{3} + x ^{2}} d x = 8 (ln ∣ x ∣ - ln ∣ x + 1 ∣ + \frac{1}{x + 1})

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

= - \frac{1}{x + 1} + 8 (ln ∣ x ∣ - ln ∣ x + 1 ∣ + \frac{1}{x + 1}) + 4 (- 2 ln ∣ x ∣ - \frac{1}{x} + 2 ln ∣ x + 1 ∣ - \frac{1}{x + 1})

Vereinfache

- \frac{1}{x + 1} + 8 (ln ∣ x ∣ - ln ∣ x + 1 ∣ + \frac{1}{x + 1}) + 4 (- 2 ln ∣ x ∣ - \frac{1}{x} + 2 ln ∣ x + 1 ∣ - \frac{1}{x + 1}) : \frac{3}{x + 1} - \frac{4}{x}

= \frac{3}{x + 1} - \frac{4}{x}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{x + 1} - \frac{4}{x} + C

Graph

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