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

Lösung

\int (2 x (x^{2} + 2)^{2}) d x

2 (\frac{x ^{6}}{6} + x^{4} + 2 x^{2}) + C

Schritte zur Lösung

\int 2 x (x^{2} + 2)^{2} d x

Entferne die Konstante:

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

= 2 \cdot \int x (x^{2} + 2)^{2} d x

Multipliziere aus

x (x^{2} + 2)^{2} : x^{5} + 4 x^{3} + 4 x

= 2 \cdot \int x^{5} + 4 x^{3} + 4 x d x

Wende die Summenregel an:

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

= 2 (\int x^{5} d x + \int 4 x^{3} d x + \int 4 x d x)

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

\int 4 x^{3} d x = x^{4}

\int 4 x d x = 2 x^{2}

= 2 (\frac{x ^{6}}{6} + x^{4} + 2 x^{2})

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{x ^{6}}{6} + x^{4} + 2 x^{2}) + C

d er i v a t i v e t sin (t) + cos (t)

2 t^{2} \frac{d x}{d t} = 3 t x + x^{2}

d er i v a t i v e y = \frac{5}{4 x ^{3}}

x \to 0 - lim (x^{2})

\frac{d N}{d t} + N = Nt e^{t + 5}