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

解

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

\frac{158}{3}

十進法表記

52.66666 \dots

解答ステップ

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

拡張

x (x^{2} + 3)^{2} : x^{5} + 6 x^{3} + 9 x

= \int_{0}^{2} x^{5} + 6 x^{3} + 9 x d x

総和規則を適用する:

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

= \int_{0}^{2} x^{5} d x + \int_{0}^{2} 6 x^{3} d x + \int_{0}^{2} 9 x d x

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

\int_{0}^{2} 6 x^{3} d x = 24

\int_{0}^{2} 9 x d x = 18

= \frac{32}{3} + 24 + 18

簡素化

\frac{32}{3} + 24 + 18 : \frac{158}{3}

= \frac{158}{3}