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Beliebt Rechnen >

integral of (x^2)/(9+x)

Lösung

\int \frac{x ^{2}}{9 + x} d x

Lösung

\frac{( 9 + x ) ^{2}}{2} - 18 (9 + x) + 81 ln ∣ 9 + x ∣ + C

Schritte zur Lösung

\int \frac{x ^{2}}{9 + x} d x

Wende U-Substitution an

= \int \frac{( u - 9 ) ^{2}}{u} d u

Multipliziere aus

\frac{( u - 9 ) ^{2}}{u} : u - 18 + \frac{81}{u}

= \int u - 18 + \frac{81}{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 18 d u + \int \frac{81}{u} d u

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

\int 18 d u = 18 u

\int \frac{81}{u} d u = 81 ln ∣ u ∣

= \frac{u ^{2}}{2} - 18 u + 81 ln ∣ u ∣

Setze in

u = 9 + x

ein

= \frac{( 9 + x ) ^{2}}{2} - 18 (9 + x) + 81 ln ∣ 9 + x ∣

Füge eine Konstante zur Lösung hinzu

= \frac{( 9 + x ) ^{2}}{2} - 18 (9 + x) + 81 ln ∣ 9 + x ∣ + C

Graph

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