integral of x^2(5x^3+7x^2-6x-3)

解

\int x^{2} (5 x^{3} + 7 x^{2} - 6 x - 3) d x

\frac{5 x ^{6}}{6} + \frac{7 x ^{5}}{5} - \frac{3 x ^{4}}{2} - x^{3} + C

解答ステップ

\int x^{2} (5 x^{3} + 7 x^{2} - 6 x - 3) d x

拡張

x^{2} (5 x^{3} + 7 x^{2} - 6 x - 3) : 5 x^{5} + 7 x^{4} - 6 x^{3} - 3 x^{2}

= \int 5 x^{5} + 7 x^{4} - 6 x^{3} - 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

= \int 5 x^{5} d x + \int 7 x^{4} d x - \int 6 x^{3} d x - \int 3 x^{2} d x

\int 5 x^{5} d x = \frac{5 x ^{6}}{6}

\int 7 x^{4} d x = \frac{7 x ^{5}}{5}

\int 6 x^{3} d x = \frac{3 x ^{4}}{2}

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

= \frac{5 x ^{6}}{6} + \frac{7 x ^{5}}{5} - \frac{3 x ^{4}}{2} - x^{3}

定数を解答に追加する

= \frac{5 x ^{6}}{6} + \frac{7 x ^{5}}{5} - \frac{3 x ^{4}}{2} - x^{3} + C