integral of 2x(4x^2-10)^2

解

\int 2 x (4 x^{2} - 10)^{2} d x

2 (\frac{8 x ^{6}}{3} - 20 x^{4} + 50 x^{2}) + C

解答ステップ

\int 2 x (4 x^{2} - 10)^{2} d x

定数を除く:

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

= 2 \cdot \int x (4 x^{2} - 10)^{2} d x

拡張

x (4 x^{2} - 10)^{2} : 16 x^{5} - 80 x^{3} + 100 x

= 2 \cdot \int 16 x^{5} - 80 x^{3} + 100 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

= 2 (\int 16 x^{5} d x - \int 80 x^{3} d x + \int 100 x d x)

\int 16 x^{5} d x = \frac{8 x ^{6}}{3}

\int 80 x^{3} d x = 20 x^{4}

\int 100 x d x = 50 x^{2}

= 2 (\frac{8 x ^{6}}{3} - 20 x^{4} + 50 x^{2})

定数を解答に追加する

= 2 (\frac{8 x ^{6}}{3} - 20 x^{4} + 50 x^{2}) + C