integral of x(x-6)^7

解答

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

\frac{1}{2} x^{2} (x - 6)^{7} - \frac{7 x ^{9}}{18} + \frac{63 x ^{8}}{4} - 270 x^{7} + 2520 x^{6} - 13608 x^{5} + 40824 x^{4} - 54432 x^{3} + C

求解步骤

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

使用分布积分法

= \frac{1}{2} x^{2} (x - 6)^{7} - \int \frac{7}{2} x^{2} (x - 6)^{6} d x

\int \frac{7}{2} x^{2} (x - 6)^{6} d x = \frac{7 x ^{9}}{18} - \frac{63 x ^{8}}{4} + 270 x^{7} - 2520 x^{6} + 13608 x^{5} - 40824 x^{4} + 54432 x^{3}

= \frac{1}{2} x^{2} (x - 6)^{7} - (\frac{7 x ^{9}}{18} - \frac{63 x ^{8}}{4} + 270 x^{7} - 2520 x^{6} + 13608 x^{5} - 40824 x^{4} + 54432 x^{3})

化简

= \frac{1}{2} x^{2} (x - 6)^{7} - \frac{7 x ^{9}}{18} + \frac{63 x ^{8}}{4} - 270 x^{7} + 2520 x^{6} - 13608 x^{5} + 40824 x^{4} - 54432 x^{3}

解答补常数

= \frac{1}{2} x^{2} (x - 6)^{7} - \frac{7 x ^{9}}{18} + \frac{63 x ^{8}}{4} - 270 x^{7} + 2520 x^{6} - 13608 x^{5} + 40824 x^{4} - 54432 x^{3} + C

\int \frac{x}{- x ^{2} - 8 x - 7} d x

\int x^{3} 3 8 - 3 x^{4} d x

\int \frac{1}{2 x - 1} d x

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

\int - 9 sin (x) d x