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

Lösung

\int \frac{x ^{2}}{x + 6} d x

\frac{( x + 6 ) ^{2}}{2} - 12 (x + 6) + 36 ln ∣ x + 6 ∣ + C

Schritte zur Lösung

\int \frac{x ^{2}}{x + 6} d x

Wende U-Substitution an

= \int \frac{( u - 6 ) ^{2}}{u} d u

Multipliziere aus

\frac{( u - 6 ) ^{2}}{u} : u - 12 + \frac{36}{u}

= \int u - 12 + \frac{36}{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

= \int u d u - \int 12 d u + \int \frac{36}{u} d u

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

\int 12 d u = 12 u

\int \frac{36}{u} d u = 36 ln ∣ u ∣

= \frac{u ^{2}}{2} - 12 u + 36 ln ∣ u ∣

Setze in

u = x + 6

ein

= \frac{( x + 6 ) ^{2}}{2} - 12 (x + 6) + 36 ln ∣ x + 6 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{( x + 6 ) ^{2}}{2} - 12 (x + 6) + 36 ln ∣ x + 6 ∣ + C

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