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Beliebt Rechnen >

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

Lösung

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

Lösung

ln ∣ x ∣ + arctan (x) + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

\int \frac{x}{x + x ^{3}} d x = arctan (x)

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

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

Vereinfache

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

= ln ∣ x ∣ + arctan (x)

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ + arctan (x) + C

Graph

Beliebte Beispiele

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