integral from 10 to 20 of 10(3x+4)^2

解

\int_{10}^{20} 10 (3 x + 4)^{2} d x

247600

解答ステップ

\int_{10}^{20} 10 (3 x + 4)^{2} d x

定数を除く:

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

= 10 \cdot \int_{10}^{20} (3 x + 4)^{2} d x

拡張

(3 x + 4)^{2} : 9 x^{2} + 24 x + 16

= 10 \cdot \int_{10}^{20} 9 x^{2} + 24 x + 16 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

= 10 (\int_{10}^{20} 9 x^{2} d x + \int_{10}^{20} 24 x d x + \int_{10}^{20} 16 d x)

\int_{10}^{20} 9 x^{2} d x = 21000

\int_{10}^{20} 24 x d x = 3600

\int_{10}^{20} 16 d x = 160

= 10 (21000 + 3600 + 160)

簡素化

= 247600