integral of x(x+3)^6

解答

\int x (x + 3)^{6} d x

\frac{x ^{8}}{8} + \frac{18 x ^{7}}{7} + \frac{45 x ^{6}}{2} + 108 x^{5} + \frac{1215 x ^{4}}{4} + 486 x^{3} + \frac{729 x ^{2}}{2} + C

求解步骤

\int x (x + 3)^{6} d x

乘开

x (x + 3)^{6} : x^{7} + 18 x^{6} + 135 x^{5} + 540 x^{4} + 1215 x^{3} + 1458 x^{2} + 729 x

= \int x^{7} + 18 x^{6} + 135 x^{5} + 540 x^{4} + 1215 x^{3} + 1458 x^{2} + 729 x d x

使用积分加法定则:

\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 18 x^{6} d x + \int 135 x^{5} d x + \int 540 x^{4} d x + \int 1215 x^{3} d x + \int 1458 x^{2} d x + \int 729 x d x

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

\int 18 x^{6} d x = \frac{18 x ^{7}}{7}

\int 135 x^{5} d x = \frac{45 x ^{6}}{2}

\int 540 x^{4} d x = 108 x^{5}

\int 1215 x^{3} d x = \frac{1215 x ^{4}}{4}

\int 1458 x^{2} d x = 486 x^{3}

\int 729 x d x = \frac{729 x ^{2}}{2}

= \frac{x ^{8}}{8} + \frac{18 x ^{7}}{7} + \frac{45 x ^{6}}{2} + 108 x^{5} + \frac{1215 x ^{4}}{4} + 486 x^{3} + \frac{729 x ^{2}}{2}

解答补常数

= \frac{x ^{8}}{8} + \frac{18 x ^{7}}{7} + \frac{45 x ^{6}}{2} + 108 x^{5} + \frac{1215 x ^{4}}{4} + 486 x^{3} + \frac{729 x ^{2}}{2} + C