integral from-x to x^2 of (8x-10y+2)

解

\int_{- x}^{x^{2}} (8 x - 10 y + 2) d y

- 5 x^{4} + 8 x^{3} + 15 x^{2} + 2 x

解答ステップ

\int_{- x}^{x^{2}} 8 x - 10 y + 2 d y

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_{- x}^{x^{2}} 2 d y - \int_{- x}^{x^{2}} 10 y d y + \int_{- x}^{x^{2}} 8 x d y

\int_{- x}^{x^{2}} 2 d y = 2 x^{2} + 2 x

\int_{- x}^{x^{2}} 10 y d y = 5 (x^{4} - x^{2})

\int_{- x}^{x^{2}} 8 x d y = 8 x^{3} + 8 x^{2}

= 2 x^{2} + 2 x - 5 (x^{4} - x^{2}) + 8 x^{3} + 8 x^{2}

簡素化

2 x^{2} + 2 x - 5 (x^{4} - x^{2}) + 8 x^{3} + 8 x^{2} : - 5 x^{4} + 8 x^{3} + 15 x^{2} + 2 x

= - 5 x^{4} + 8 x^{3} + 15 x^{2} + 2 x