integral from 2x^2 to 1+x^2 of x+2y

解

\int_{2 x^{2}}^{1 + x^{2}} x + 2 y d y

x - x^{3} + 1 + 2 x^{2} - 3 x^{4}

解答ステップ

\int_{2 x^{2}}^{1 + x^{2}} x + 2 y 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_{2 x^{2}}^{1 + x^{2}} x d y + \int_{2 x^{2}}^{1 + x^{2}} 2 y d y

\int_{2 x^{2}}^{1 + x^{2}} x d y = x - x^{3}

\int_{2 x^{2}}^{1 + x^{2}} 2 y d y = 1 + 2 x^{2} - 3 x^{4}

= x - x^{3} + 1 + 2 x^{2} - 3 x^{4}