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

解

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

3 (\frac{x ^{5}}{5} + x^{3}) + C

解答ステップ

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

定数を除く:

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

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

拡張

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

= 3 \cdot \int x^{4} + 3 x^{2} 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 x^{4} d x + \int 3 x^{2} d x)

\int x^{4} d x = \frac{x ^{5}}{5}

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

= 3 (\frac{x ^{5}}{5} + x^{3})

定数を解答に追加する

= 3 (\frac{x ^{5}}{5} + x^{3}) + C