integral from-x to x^2 of (8x-10y+4)

解

\int_{- x}^{x^{2}} (8 x - 10 y + 4) d x

4 x^{4} - 10 y x^{2} - 10 y x + 4 x

解答ステップ

\int_{- x}^{x^{2}} 8 x - 10 y + 4 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_{- x}^{x^{2}} 4 d x - \int_{- x}^{x^{2}} 10 y d x + \int_{- x}^{x^{2}} 8 x d x

\int_{- x}^{x^{2}} 4 d x = 4 x^{2} + 4 x

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

\int_{- x}^{x^{2}} 8 x d x = 4 (x^{4} - x^{2})

= 4 x^{2} + 4 x - (10 y x^{2} + 10 y x) + 4 (x^{4} - x^{2})

簡素化

4 x^{2} + 4 x - (10 y x^{2} + 10 y x) + 4 (x^{4} - x^{2}) : 4 x^{4} - 10 y x^{2} - 10 y x + 4 x

= 4 x^{4} - 10 y x^{2} - 10 y x + 4 x