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Beliebt Rechnen >

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

Lösung

\int \frac{x ( x ^{2} - 1 )}{x ^{2} + 1} d x

Lösung

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

Schritte zur Lösung

\int \frac{x ( x ^{2} - 1 )}{x ^{2} + 1} d x

Wende U-Substitution an

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

\frac{u - 1}{u + 1} = - \frac{2}{u + 1} + 1

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

\int \frac{2}{u + 1} d u = 2 ln ∣ u + 1 ∣

\int 1 d u = u

= \frac{1}{2} (- 2 ln ∣ u + 1 ∣ + u)

Setze in

u = x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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