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Beliebt Rechnen >

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

Lösung

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

Lösung

- ln ∣ x + 2 ∣ + \frac{1}{2} ln x^{2} + 1 + arctan (x) + C

Schritte zur Lösung

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

Multipliziere aus

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

\int \frac{3 x}{x ^{3} + 2 x ^{2} + x + 2} d x = 3 (- \frac{2}{5} ln ∣ x + 2 ∣ + \frac{1}{5} (ln x^{2} + 1 + arctan (x)))

\int \frac{1}{x ^{3} + 2 x ^{2} + x + 2} d x = \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} (- \frac{1}{2} ln x^{2} + 1 + 2 arctan (x))

= 3 (- \frac{2}{5} ln ∣ x + 2 ∣ + \frac{1}{5} (ln x^{2} + 1 + arctan (x))) + \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} (- \frac{1}{2} ln x^{2} + 1 + 2 arctan (x))

Vereinfache

3 (- \frac{2}{5} ln ∣ x + 2 ∣ + \frac{1}{5} (ln x^{2} + 1 + arctan (x))) + \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} (- \frac{1}{2} ln x^{2} + 1 + 2 arctan (x)) : - ln ∣ x + 2 ∣ + \frac{1}{2} ln x^{2} + 1 + arctan (x)

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

Füge eine Konstante zur Lösung hinzu

= - ln ∣ x + 2 ∣ + \frac{1}{2} ln x^{2} + 1 + arctan (x) + C

Graph

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