integral of (4x-6)sqrt(x^2-3x)

Lösung

\int (4 x - 6) x^{2} - 3 x d x

\frac{4}{3} (x (x - 3))^{\frac{3}{2}} + C

Schritte zur Lösung

\int (4 x - 6) x^{2} - 3 x d x

Wende U-Substitution an

= \int \frac{u u ^{2} - 36}{16} d u

Entferne die Konstante:

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

= \frac{1}{16} \cdot \int u u^{2} - 36 d u

Wende U-Substitution an

= \frac{1}{16} \cdot \int v^{2} d v

Wende die Potenzregel an

= \frac{1}{16} \cdot \frac{v ^{3}}{3}

Ersetze zurück

= \frac{1}{16} \cdot \frac{( ( 4 x - 6 ) ^{2} - 36 ) ^{3}}{3}

Vereinfache

\frac{1}{16} \cdot \frac{( ( 4 x - 6 ) ^{2} - 36 ) ^{3}}{3} : \frac{4}{3} (x (x - 3))^{\frac{3}{2}}

= \frac{4}{3} (x (x - 3))^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{4}{3} (x (x - 3))^{\frac{3}{2}} + C

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