integral from 1 to 3 of 5x(x^2+1)^3

解

\int_{1}^{3} 5 x (x^{2} + 1)^{3} d x

6240

解答ステップ

\int_{1}^{3} 5 x (x^{2} + 1)^{3} d x

定数を除く:

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

= 5 \cdot \int_{1}^{3} x (x^{2} + 1)^{3} d x

拡張

x (x^{2} + 1)^{3} : x^{7} + 3 x^{5} + 3 x^{3} + x

= 5 \cdot \int_{1}^{3} x^{7} + 3 x^{5} + 3 x^{3} + 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

= 5 (\int_{1}^{3} x^{7} d x + \int_{1}^{3} 3 x^{5} d x + \int_{1}^{3} 3 x^{3} d x + \int_{1}^{3} x d x)

\int_{1}^{3} x^{7} d x = 820

\int_{1}^{3} 3 x^{5} d x = 364

\int_{1}^{3} 3 x^{3} d x = 60

\int_{1}^{3} x d x = 4

= 5 (820 + 364 + 60 + 4)

簡素化

= 6240