integral of 36(x^2+1)(x^3+3x)^2

解

\int 36 (x^{2} + 1) (x^{3} + 3 x)^{2} d x

36 (\frac{x ^{9}}{9} + x^{7} + 3 x^{5} + 3 x^{3}) + C

解答ステップ

\int 36 (x^{2} + 1) (x^{3} + 3 x)^{2} d x

定数を除く:

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

= 36 \cdot \int (x^{2} + 1) (x^{3} + 3 x)^{2} d x

拡張

(x^{2} + 1) (x^{3} + 3 x)^{2} : x^{8} + 7 x^{6} + 15 x^{4} + 9 x^{2}

= 36 \cdot \int x^{8} + 7 x^{6} + 15 x^{4} + 9 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

= 36 (\int x^{8} d x + \int 7 x^{6} d x + \int 15 x^{4} d x + \int 9 x^{2} d x)

\int x^{8} d x = \frac{x ^{9}}{9}

\int 7 x^{6} d x = x^{7}

\int 15 x^{4} d x = 3 x^{5}

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

= 36 (\frac{x ^{9}}{9} + x^{7} + 3 x^{5} + 3 x^{3})

定数を解答に追加する

= 36 (\frac{x ^{9}}{9} + x^{7} + 3 x^{5} + 3 x^{3}) + C