integral from 0 to 2 of 4x(x^2+1)^3

Lösung

\int_{0}^{2} 4 x (x^{2} + 1)^{3} d x

312

Schritte zur Lösung

\int_{0}^{2} 4 x (x^{2} + 1)^{3} d x

Entferne die Konstante:

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

= 4 \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

= 4 \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

= 4 (\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

= 4 (32 + 32 + 12 + 2)

Vereinfache

= 312

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