integral of x^2(3x^2+9)^2

解

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

\frac{9 x ^{7}}{7} + \frac{54 x ^{5}}{5} + 27 x^{3} + C

解答ステップ

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

拡張

x^{2} (3 x^{2} + 9)^{2} : 9 x^{6} + 54 x^{4} + 81 x^{2}

= \int 9 x^{6} + 54 x^{4} + 81 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

= \int 9 x^{6} d x + \int 54 x^{4} d x + \int 81 x^{2} d x

\int 9 x^{6} d x = \frac{9 x ^{7}}{7}

\int 54 x^{4} d x = \frac{54 x ^{5}}{5}

\int 81 x^{2} d x = 27 x^{3}

= \frac{9 x ^{7}}{7} + \frac{54 x ^{5}}{5} + 27 x^{3}

定数を解答に追加する

= \frac{9 x ^{7}}{7} + \frac{54 x ^{5}}{5} + 27 x^{3} + C