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Beliebt Rechnen >

integral of (x^2)/(7+x)

Lösung

\int \frac{x ^{2}}{7 + x} d x

Lösung

\frac{( 7 + x ) ^{2}}{2} - 14 (7 + x) + 49 ln ∣ 7 + x ∣ + C

Schritte zur Lösung

\int \frac{x ^{2}}{7 + x} d x

Wende U-Substitution an

= \int \frac{( u - 7 ) ^{2}}{u} d u

Multipliziere aus

\frac{( u - 7 ) ^{2}}{u} : u - 14 + \frac{49}{u}

= \int u - 14 + \frac{49}{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 14 d u + \int \frac{49}{u} d u

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

\int 14 d u = 14 u

\int \frac{49}{u} d u = 49 ln ∣ u ∣

= \frac{u ^{2}}{2} - 14 u + 49 ln ∣ u ∣

Setze in

u = 7 + x

ein

= \frac{( 7 + x ) ^{2}}{2} - 14 (7 + x) + 49 ln ∣ 7 + x ∣

Füge eine Konstante zur Lösung hinzu

= \frac{( 7 + x ) ^{2}}{2} - 14 (7 + x) + 49 ln ∣ 7 + x ∣ + C

Graph

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