integral of x(x+7)^6

解

\int x (x + 7)^{6} d x

\frac{x ^{8}}{8} + 6 x^{7} + \frac{245 x ^{6}}{2} + 1372 x^{5} + \frac{36015 x ^{4}}{4} + 33614 x^{3} + \frac{117649 x ^{2}}{2} + C

解答ステップ

\int x (x + 7)^{6} d x

拡張

x (x + 7)^{6} : x^{7} + 42 x^{6} + 735 x^{5} + 6860 x^{4} + 36015 x^{3} + 100842 x^{2} + 117649 x

= \int x^{7} + 42 x^{6} + 735 x^{5} + 6860 x^{4} + 36015 x^{3} + 100842 x^{2} + 117649 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^{7} d x + \int 42 x^{6} d x + \int 735 x^{5} d x + \int 6860 x^{4} d x + \int 36015 x^{3} d x + \int 100842 x^{2} d x + \int 117649 x d x

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

\int 42 x^{6} d x = 6 x^{7}

\int 735 x^{5} d x = \frac{245 x ^{6}}{2}

\int 6860 x^{4} d x = 1372 x^{5}

\int 36015 x^{3} d x = \frac{36015 x ^{4}}{4}

\int 100842 x^{2} d x = 33614 x^{3}

\int 117649 x d x = \frac{117649 x ^{2}}{2}

= \frac{x ^{8}}{8} + 6 x^{7} + \frac{245 x ^{6}}{2} + 1372 x^{5} + \frac{36015 x ^{4}}{4} + 33614 x^{3} + \frac{117649 x ^{2}}{2}

定数を解答に追加する

= \frac{x ^{8}}{8} + 6 x^{7} + \frac{245 x ^{6}}{2} + 1372 x^{5} + \frac{36015 x ^{4}}{4} + 33614 x^{3} + \frac{117649 x ^{2}}{2} + C