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Beliebt Rechnen >

integral of (x^4+1)/(x^3+2x)

Lösung

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

Lösung

\frac{1}{2} x^{2} + \frac{1}{2} ln ∣ x ∣ - \frac{5}{4} ln x^{2} + 2 + 1 + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

= \frac{1}{2} x^{2} + 1 - ln x^{2} + 2 + \frac{1}{2} ln ∣ x ∣ - \frac{1}{4} ln x^{2} + 2

Vereinfache

\frac{1}{2} x^{2} + 1 - ln x^{2} + 2 + \frac{1}{2} ln ∣ x ∣ - \frac{1}{4} ln x^{2} + 2 : \frac{1}{2} x^{2} + \frac{1}{2} ln ∣ x ∣ - \frac{5}{4} ln x^{2} + 2 + 1

= \frac{1}{2} x^{2} + \frac{1}{2} ln ∣ x ∣ - \frac{5}{4} ln x^{2} + 2 + 1

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x^{2} + \frac{1}{2} ln ∣ x ∣ - \frac{5}{4} ln x^{2} + 2 + 1 + C

Graph

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