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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{4 x ^{3}}{x ^{2} + 1} + \frac{4 x}{x ^{2} + 1} + 4 ln ∣ x ∣ - 2 ln x^{2} + 1 + C

Schritte zur Lösung

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

Multipliziere aus

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

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

\int \frac{4 x}{x ^{3} + x} d x = 4 arctan (x)

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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