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Beliebt Rechnen >

integral of (3x^3)/(x-2)

Lösung

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

Lösung

9 x^{2} + 24 ln ∣ x - 2 ∣ + (x - 2)^{3} - 36 + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende U-Substitution an

= 3 \cdot \int \frac{( u + 2 ) ^{3}}{u} d u

Multipliziere aus

\frac{( u + 2 ) ^{3}}{u} : u^{2} + 6 u + 12 + \frac{8}{u}

= 3 \cdot \int u^{2} + 6 u + 12 + \frac{8}{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

= 3 (\int u^{2} d u + \int 6 u d u + \int 12 d u + \int \frac{8}{u} d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int 6 u d u = 3 u^{2}

\int 12 d u = 12 u

\int \frac{8}{u} d u = 8 ln ∣ u ∣

= 3 (\frac{u ^{3}}{3} + 3 u^{2} + 12 u + 8 ln ∣ u ∣)

Setze in

u = x - 2

ein

= 3 (\frac{( x - 2 ) ^{3}}{3} + 3 (x - 2)^{2} + 12 (x - 2) + 8 ln ∣ x - 2 ∣)

Vereinfache

3 (\frac{( x - 2 ) ^{3}}{3} + 3 (x - 2)^{2} + 12 (x - 2) + 8 ln ∣ x - 2 ∣) : 9 x^{2} + 24 ln ∣ x - 2 ∣ + (x - 2)^{3} - 36

= 9 x^{2} + 24 ln ∣ x - 2 ∣ + (x - 2)^{3} - 36

Füge eine Konstante zur Lösung hinzu

= 9 x^{2} + 24 ln ∣ x - 2 ∣ + (x - 2)^{3} - 36 + C

Graph

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