integral of 7v(v^2+6)^2

Lösung

\int 7 v (v^{2} + 6)^{2} d v

7 (\frac{v ^{6}}{6} + 3 v^{4} + 18 v^{2}) + C

Schritte zur Lösung

\int 7 v (v^{2} + 6)^{2} d v

Entferne die Konstante:

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

= 7 \cdot \int v (v^{2} + 6)^{2} d v

Multipliziere aus

v (v^{2} + 6)^{2} : v^{5} + 12 v^{3} + 36 v

= 7 \cdot \int v^{5} + 12 v^{3} + 36 v d v

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 7 (\int v^{5} d v + \int 12 v^{3} d v + \int 36 v d v)

\int v^{5} d v = \frac{v ^{6}}{6}

\int 12 v^{3} d v = 3 v^{4}

\int 36 v d v = 18 v^{2}

= 7 (\frac{v ^{6}}{6} + 3 v^{4} + 18 v^{2})

Füge eine Konstante zur Lösung hinzu

= 7 (\frac{v ^{6}}{6} + 3 v^{4} + 18 v^{2}) + C

a re a 6 x - x^{3}, ∣ 2 \cdot x ∣

d er i v a t i v e ln (16 + x^{2})

v \to 1 lim (\frac{v ^{2} - 5}{v})

x \to \infty + lim (3 \cdot + 2^{x})

\int \frac{- 7 x}{4 x + 2} d x