integral from 0 to 2 of (x^4+5)^3x^3

解

\int_{0}^{2} (x^{4} + 5)^{3} x^{3} d x

12116

解答ステップ

\int_{0}^{2} (x^{4} + 5)^{3} x^{3} d x

拡張

(x^{4} + 5)^{3} x^{3} : x^{15} + 15 x^{11} + 75 x^{7} + 125 x^{3}

= \int_{0}^{2} x^{15} + 15 x^{11} + 75 x^{7} + 125 x^{3} 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^{15} d x + \int_{0}^{2} 15 x^{11} d x + \int_{0}^{2} 75 x^{7} d x + \int_{0}^{2} 125 x^{3} d x

\int_{0}^{2} x^{15} d x = 4096

\int_{0}^{2} 15 x^{11} d x = 5120

\int_{0}^{2} 75 x^{7} d x = 2400

\int_{0}^{2} 125 x^{3} d x = 500

= 4096 + 5120 + 2400 + 500

簡素化

= 12116