integral of x(x+9)^5

解

\int x (x + 9)^{5} d x

\frac{x ^{7}}{7} + \frac{15 x ^{6}}{2} + 162 x^{5} + \frac{3645 x ^{4}}{2} + 10935 x^{3} + \frac{59049 x ^{2}}{2} + C

解答ステップ

\int x (x + 9)^{5} d x

拡張

x (x + 9)^{5} : x^{6} + 45 x^{5} + 810 x^{4} + 7290 x^{3} + 32805 x^{2} + 59049 x

= \int x^{6} + 45 x^{5} + 810 x^{4} + 7290 x^{3} + 32805 x^{2} + 59049 x 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 x^{6} d x + \int 45 x^{5} d x + \int 810 x^{4} d x + \int 7290 x^{3} d x + \int 32805 x^{2} d x + \int 59049 x d x

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

\int 45 x^{5} d x = \frac{15 x ^{6}}{2}

\int 810 x^{4} d x = 162 x^{5}

\int 7290 x^{3} d x = \frac{3645 x ^{4}}{2}

\int 32805 x^{2} d x = 10935 x^{3}

\int 59049 x d x = \frac{59049 x ^{2}}{2}

= \frac{x ^{7}}{7} + \frac{15 x ^{6}}{2} + 162 x^{5} + \frac{3645 x ^{4}}{2} + 10935 x^{3} + \frac{59049 x ^{2}}{2}

定数を解答に追加する

= \frac{x ^{7}}{7} + \frac{15 x ^{6}}{2} + 162 x^{5} + \frac{3645 x ^{4}}{2} + 10935 x^{3} + \frac{59049 x ^{2}}{2} + C