integral of (x+7)^7(x+8)

解

\int (x + 7)^{7} (x + 8) d x

\frac{x ^{9}}{9} + \frac{57 x ^{8}}{8} + 203 x^{7} + \frac{20237 x ^{6}}{6} + 36015 x^{5} + \frac{1025227 x ^{4}}{4} + \frac{3647119 x ^{3}}{3} + \frac{7411887 x ^{2}}{2} + 6588344 x + C

解答ステップ

\int (x + 7)^{7} (x + 8) d x

拡張

(x + 7)^{7} (x + 8) : x^{8} + 57 x^{7} + 1421 x^{6} + 20237 x^{5} + 180075 x^{4} + 1025227 x^{3} + 3647119 x^{2} + 7411887 x + 6588344

= \int x^{8} + 57 x^{7} + 1421 x^{6} + 20237 x^{5} + 180075 x^{4} + 1025227 x^{3} + 3647119 x^{2} + 7411887 x + 6588344 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^{8} d x + \int 57 x^{7} d x + \int 1421 x^{6} d x + \int 20237 x^{5} d x + \int 180075 x^{4} d x + \int 1025227 x^{3} d x + \int 3647119 x^{2} d x + \int 7411887 x d x + \int 6588344 d x

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

\int 57 x^{7} d x = \frac{57 x ^{8}}{8}

\int 1421 x^{6} d x = 203 x^{7}

\int 20237 x^{5} d x = \frac{20237 x ^{6}}{6}

\int 180075 x^{4} d x = 36015 x^{5}

\int 1025227 x^{3} d x = \frac{1025227 x ^{4}}{4}

\int 3647119 x^{2} d x = \frac{3647119 x ^{3}}{3}

\int 7411887 x d x = \frac{7411887 x ^{2}}{2}

\int 6588344 d x = 6588344 x

= \frac{x ^{9}}{9} + \frac{57 x ^{8}}{8} + 203 x^{7} + \frac{20237 x ^{6}}{6} + 36015 x^{5} + \frac{1025227 x ^{4}}{4} + \frac{3647119 x ^{3}}{3} + \frac{7411887 x ^{2}}{2} + 6588344 x

定数を解答に追加する

= \frac{x ^{9}}{9} + \frac{57 x ^{8}}{8} + 203 x^{7} + \frac{20237 x ^{6}}{6} + 36015 x^{5} + \frac{1025227 x ^{4}}{4} + \frac{3647119 x ^{3}}{3} + \frac{7411887 x ^{2}}{2} + 6588344 x + C