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Beliebt Rechnen >

integral of (5r^2)/(r+7)

Lösung

\int \frac{5 r ^{2}}{r + 7} d r

Lösung

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

Schritte zur Lösung

\int \frac{5 r ^{2}}{r + 7} d r

Entferne die Konstante:

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

= 5 \cdot \int \frac{r ^{2}}{r + 7} d r

Wende U-Substitution an

= 5 \cdot \int \frac{( u - 7 ) ^{2}}{u} d u

Multipliziere aus

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

= 5 \cdot \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

= 5 (\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 ∣

= 5 (\frac{u ^{2}}{2} - 14 u + 49 ln ∣ u ∣)

Setze in

u = r + 7

ein

= 5 (\frac{( r + 7 ) ^{2}}{2} - 14 (r + 7) + 49 ln ∣ r + 7 ∣)

Füge eine Konstante zur Lösung hinzu

= 5 (\frac{( r + 7 ) ^{2}}{2} - 14 (r + 7) + 49 ln ∣ r + 7 ∣) + C

Graph

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