integral of x(x+7)^4

解

\int x (x + 7)^{4} d x

\frac{x ^{6}}{6} + \frac{28 x ^{5}}{5} + \frac{147 x ^{4}}{2} + \frac{1372 x ^{3}}{3} + \frac{2401 x ^{2}}{2} + C

解答ステップ

\int x (x + 7)^{4} d x

拡張

x (x + 7)^{4} : x^{5} + 28 x^{4} + 294 x^{3} + 1372 x^{2} + 2401 x

= \int x^{5} + 28 x^{4} + 294 x^{3} + 1372 x^{2} + 2401 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^{5} d x + \int 28 x^{4} d x + \int 294 x^{3} d x + \int 1372 x^{2} d x + \int 2401 x d x

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

\int 28 x^{4} d x = \frac{28 x ^{5}}{5}

\int 294 x^{3} d x = \frac{147 x ^{4}}{2}

\int 1372 x^{2} d x = \frac{1372 x ^{3}}{3}

\int 2401 x d x = \frac{2401 x ^{2}}{2}

= \frac{x ^{6}}{6} + \frac{28 x ^{5}}{5} + \frac{147 x ^{4}}{2} + \frac{1372 x ^{3}}{3} + \frac{2401 x ^{2}}{2}

定数を解答に追加する

= \frac{x ^{6}}{6} + \frac{28 x ^{5}}{5} + \frac{147 x ^{4}}{2} + \frac{1372 x ^{3}}{3} + \frac{2401 x ^{2}}{2} + C