integral from 0 to 2 of 2pix(3x(x-2)^2)

Lösung

\int_{0}^{2} 2 π x (3 x (x - 2)^{2}) d x

\frac{32 π}{5}

Dezimale

20.10619 \dots

Schritte zur Lösung

\int_{0}^{2} 2 π x \cdot 3 x (x - 2)^{2} d x

Entferne die Konstante:

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

= 2 π 3 \cdot \int_{0}^{2} xx (x - 2)^{2} d x

Vereinfache

= 6 π \cdot \int_{0}^{2} xx (x - 2)^{2} d x

Multipliziere aus

xx (x - 2)^{2} : x^{4} - 4 x^{3} + 4 x^{2}

= 6 π \cdot \int_{0}^{2} x^{4} - 4 x^{3} + 4 x^{2} 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

= 6 π (\int_{0}^{2} x^{4} d x - \int_{0}^{2} 4 x^{3} d x + \int_{0}^{2} 4 x^{2} d x)

\int_{0}^{2} x^{4} d x = \frac{32}{5}

\int_{0}^{2} 4 x^{3} d x = 16

\int_{0}^{2} 4 x^{2} d x = \frac{32}{3}

= 6 π (\frac{32}{5} - 16 + \frac{32}{3})

Vereinfache

6 π (\frac{32}{5} - 16 + \frac{32}{3}) : \frac{32 π}{5}

= \frac{32 π}{5}

n = 1 \sum \infty \frac{2 n}{3 n - 1}

\frac{d}{d x} (\frac{3 a}{4 x ^{5}})

\frac{\partial}{\partial y} (x^{6 y})

\int_{- 1}^{1} (x^{2} - x - 1) d x

\int t^{3} e^{- s t} d t