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Beliebt Rechnen >

integral of (r^2)/(r+4)

Lösung

\int \frac{r ^{2}}{r + 4} d r

Lösung

\frac{( r + 4 ) ^{2}}{2} - 8 (r + 4) + 16 ln ∣ r + 4 ∣ + C

Schritte zur Lösung

\int \frac{r ^{2}}{r + 4} d r

Wende U-Substitution an

= \int \frac{( u - 4 ) ^{2}}{u} d u

Multipliziere aus

\frac{( u - 4 ) ^{2}}{u} : u - 8 + \frac{16}{u}

= \int u - 8 + \frac{16}{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 8 d u + \int \frac{16}{u} d u

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

\int 8 d u = 8 u

\int \frac{16}{u} d u = 16 ln ∣ u ∣

= \frac{u ^{2}}{2} - 8 u + 16 ln ∣ u ∣

Setze in

u = r + 4

ein

= \frac{( r + 4 ) ^{2}}{2} - 8 (r + 4) + 16 ln ∣ r + 4 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{( r + 4 ) ^{2}}{2} - 8 (r + 4) + 16 ln ∣ r + 4 ∣ + C

Graph

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