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Beliebt Rechnen >

integral of (x^2-2x+6)(2x-2)

Lösung

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

Lösung

\frac{1}{2} (x^{2} - 2 x + 6)^{2} + C

Schritte zur Lösung

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

(x^{2} - 2 x + 6) (2 x - 2) = 2 (x^{2} - 2 x + 6) (x - 1)

= \int 2 (x^{2} - 2 x + 6) (x - 1) d x

Entferne die Konstante:

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

= 2 \cdot \int (x^{2} - 2 x + 6) (x - 1) d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die Potenzregel an

= 2 \cdot \frac{1}{2} \cdot \frac{u ^{2}}{2}

Setze in

u = x^{2} - 2 x + 6

ein

= 2 \cdot \frac{1}{2} \cdot \frac{( x ^{2} - 2 x + 6 ) ^{2}}{2}

Vereinfache

2 \cdot \frac{1}{2} \cdot \frac{( x ^{2} - 2 x + 6 ) ^{2}}{2} : \frac{1}{2} (x^{2} - 2 x + 6)^{2}

= \frac{1}{2} (x^{2} - 2 x + 6)^{2}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (x^{2} - 2 x + 6)^{2} + C

Graph

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