integral from 0 to 2 of 2x(x^2-1)^3

Lösung

\int_{0}^{2} 2 x (x^{2} - 1)^{3} d x

20

Schritte zur Lösung

\int_{0}^{2} 2 x (x^{2} - 1)^{3} d x

Entferne die Konstante:

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

= 2 \cdot \int_{0}^{2} x (x^{2} - 1)^{3} d x

Multipliziere aus

x (x^{2} - 1)^{3} : x^{7} - 3 x^{5} + 3 x^{3} - x

= 2 \cdot \int_{0}^{2} x^{7} - 3 x^{5} + 3 x^{3} - 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_{0}^{2} x^{7} d x - \int_{0}^{2} 3 x^{5} d x + \int_{0}^{2} 3 x^{3} d x - \int_{0}^{2} x d x)

\int_{0}^{2} x^{7} d x = 32

\int_{0}^{2} 3 x^{5} d x = 32

\int_{0}^{2} 3 x^{3} d x = 12

\int_{0}^{2} x d x = 2

= 2 (32 - 32 + 12 - 2)

Vereinfache

= 20

\int_{0}^{2} 3 x^{3} d x

\int_{0}^{1} 3^{x} e^{x} d x

\int_{1}^{x^{2}} e^{- t^{2}} d t

\int_{- 4}^{1} ∣ x + 2 ∣ d x

\int_{- 1}^{2} e^{3 x} d x