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

解

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

- \frac{9076}{35}

十進法表記

- 259.31428 \dots

解答ステップ

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

拡張

(x - 3 x^{2})^{3} : x^{3} - 9 x^{4} + 27 x^{5} - 27 x^{6}

= \int_{0}^{2} x^{3} - 9 x^{4} + 27 x^{5} - 27 x^{6} 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^{3} d x - \int_{0}^{2} 9 x^{4} d x + \int_{0}^{2} 27 x^{5} d x - \int_{0}^{2} 27 x^{6} d x

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

\int_{0}^{2} 9 x^{4} d x = \frac{288}{5}

\int_{0}^{2} 27 x^{5} d x = 288

\int_{0}^{2} 27 x^{6} d x = \frac{3456}{7}

= 4 - \frac{288}{5} + 288 - \frac{3456}{7}

簡素化

4 - \frac{288}{5} + 288 - \frac{3456}{7} : - \frac{9076}{35}

= - \frac{9076}{35}