integral of |x|-(x^2-2)

Lösung

\int ∣ x ∣ - (x^{2} - 2) d x

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

Schritte zur Lösung

\int ∣ x ∣ - (x^{2} - 2) d 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 ∣ x ∣ d x - \int x^{2} - 2 d x

\int ∣ x ∣ d x = \frac{x ∣ x ∣}{2}

\int x^{2} - 2 d x = \frac{x ^{3}}{3} - 2 x

= \frac{x ∣ x ∣}{2} - (\frac{x ^{3}}{3} - 2 x)

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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

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