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

解

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

\frac{64}{15}

十進法表記

4.26666 \dots

解答ステップ

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

簡素化

(2 x)^{2} - (x^{2})^{2} : 4 x^{2} - x^{4}

= \int_{0}^{2} 4 x^{2} - x^{4} 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} 4 x^{2} d x - \int_{0}^{2} x^{4} d x

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

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

= \frac{32}{3} - \frac{32}{5}

簡素化

\frac{32}{3} - \frac{32}{5} : \frac{64}{15}

= \frac{64}{15}