integral of (3x+5)^2(3)

解

\int (3 x + 5)^{2} (3) d x

3 (3 x^{3} + 15 x^{2} + 25 x) + C

解答ステップ

\int (3 x + 5)^{2} \cdot 3 d x

定数を除く:

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

= 3 \cdot \int (3 x + 5)^{2} d x

拡張

(3 x + 5)^{2} : 9 x^{2} + 30 x + 25

= 3 \cdot \int 9 x^{2} + 30 x + 25 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

= 3 (\int 9 x^{2} d x + \int 30 x d x + \int 25 d x)

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

\int 30 x d x = 15 x^{2}

\int 25 d x = 25 x

= 3 (3 x^{3} + 15 x^{2} + 25 x)

定数を解答に追加する

= 3 (3 x^{3} + 15 x^{2} + 25 x) + C