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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{2} x^{2} + \frac{1}{2} + ln x^{2} + 1 + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{u + 2}{2 u} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{u + 2}{u} d u

Multipliziere aus

\frac{u + 2}{u} : 1 + \frac{2}{u}

= \frac{1}{2} \cdot \int 1 + \frac{2}{u} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int 1 d u + \int \frac{2}{u} d u)

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= \frac{1}{2} (u + 2 ln ∣ u ∣)

Setze in

u = x^{2} + 1

ein

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

Vereinfache

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

= \frac{1}{2} x^{2} + \frac{1}{2} + ln x^{2} + 1

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x^{2} + \frac{1}{2} + ln x^{2} + 1 + C

Graph

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