integral of 18(3x+2)^2

解

\int 18 (3 x + 2)^{2} d x

18 (3 x^{3} + 6 x^{2} + 4 x) + C

解答ステップ

\int 18 (3 x + 2)^{2} d x

定数を除く:

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

= 18 \cdot \int (3 x + 2)^{2} d x

拡張

(3 x + 2)^{2} : 9 x^{2} + 12 x + 4

= 18 \cdot \int 9 x^{2} + 12 x + 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

= 18 (\int 9 x^{2} d x + \int 12 x d x + \int 4 d x)

\int 9 x^{2} d x = 3 x^{3}

\int 12 x d x = 6 x^{2}

\int 4 d x = 4 x

= 18 (3 x^{3} + 6 x^{2} + 4 x)

定数を解答に追加する

= 18 (3 x^{3} + 6 x^{2} + 4 x) + C