integral of (x+1)/(x^3+3x^2+7x+5)

Lösung

\int \frac{x + 1}{x ^{3} + 3 x ^{2} + 7 x + 5} d x

\frac{1}{8} ln \frac{1}{4} (x + 1)^{2} + 1 + \frac{1}{2} arctan (\frac{1}{2} (x + 1)) - \frac{1}{8} ln x^{2} + 2 x + 5 + C

Schritte zur Lösung

\int \frac{x + 1}{x ^{3} + 3 x ^{2} + 7 x + 5} d x

Multipliziere aus

\frac{x + 1}{x ^{3} + 3 x ^{2} + 7 x + 5} : \frac{x}{x ^{3} + 3 x ^{2} + 7 x + 5} + \frac{1}{x ^{3} + 3 x ^{2} + 7 x + 5}

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}{x ^{3} + 3 x ^{2} + 7 x + 5} d x + \int \frac{1}{x ^{3} + 3 x ^{2} + 7 x + 5} d x

\int \frac{x}{x ^{3} + 3 x ^{2} + 7 x + 5} d x = - \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{4} (\frac{1}{2} ln \frac{1}{4} (x + 1)^{2} + 1 + 2 arctan (\frac{1}{2} (x + 1)))

\int \frac{1}{x ^{3} + 3 x ^{2} + 7 x + 5} d x = \frac{1}{4} ln ∣ x + 1 ∣ - \frac{1}{8} ln x^{2} + 2 x + 5

= - \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{4} (\frac{1}{2} ln \frac{1}{4} (x + 1)^{2} + 1 + 2 arctan (\frac{1}{2} (x + 1))) + \frac{1}{4} ln ∣ x + 1 ∣ - \frac{1}{8} ln x^{2} + 2 x + 5

Vereinfache

- \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{4} (\frac{1}{2} ln \frac{1}{4} (x + 1)^{2} + 1 + 2 arctan (\frac{1}{2} (x + 1))) + \frac{1}{4} ln ∣ x + 1 ∣ - \frac{1}{8} ln x^{2} + 2 x + 5 : \frac{1}{8} ln \frac{1}{4} (x + 1)^{2} + 1 + \frac{1}{2} arctan (\frac{1}{2} (x + 1)) - \frac{1}{8} ln x^{2} + 2 x + 5

= \frac{1}{8} ln \frac{1}{4} (x + 1)^{2} + 1 + \frac{1}{2} arctan (\frac{1}{2} (x + 1)) - \frac{1}{8} ln x^{2} + 2 x + 5

Füge eine Konstante zur Lösung hinzu

= \frac{1}{8} ln \frac{1}{4} (x + 1)^{2} + 1 + \frac{1}{2} arctan (\frac{1}{2} (x + 1)) - \frac{1}{8} ln x^{2} + 2 x + 5 + C

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