integral of x(x-3)^7

解答

\int x (x - 3)^{7} d x

\frac{1}{2} x^{2} (x - 3)^{7} - \frac{7 x ^{9}}{18} + \frac{63 x ^{8}}{8} - \frac{135 x ^{7}}{2} + 315 x^{6} - \frac{1701}{2} x^{5} + \frac{5103 x ^{4}}{4} - \frac{1701}{2} x^{3} + C

求解步骤

\int x (x - 3)^{7} d x

使用分布积分法

= \frac{1}{2} x^{2} (x - 3)^{7} - \int \frac{7}{2} x^{2} (x - 3)^{6} d x

\int \frac{7}{2} x^{2} (x - 3)^{6} d x = \frac{7 x ^{9}}{18} - \frac{63 x ^{8}}{8} + \frac{135 x ^{7}}{2} - 315 x^{6} + \frac{1701}{2} x^{5} - \frac{5103 x ^{4}}{4} + \frac{1701}{2} x^{3}

= \frac{1}{2} x^{2} (x - 3)^{7} - (\frac{7 x ^{9}}{18} - \frac{63 x ^{8}}{8} + \frac{135 x ^{7}}{2} - 315 x^{6} + \frac{1701}{2} x^{5} - \frac{5103 x ^{4}}{4} + \frac{1701}{2} x^{3})

化简

= \frac{1}{2} x^{2} (x - 3)^{7} - \frac{7 x ^{9}}{18} + \frac{63 x ^{8}}{8} - \frac{135 x ^{7}}{2} + 315 x^{6} - \frac{1701}{2} x^{5} + \frac{5103 x ^{4}}{4} - \frac{1701}{2} x^{3}

解答补常数

= \frac{1}{2} x^{2} (x - 3)^{7} - \frac{7 x ^{9}}{18} + \frac{63 x ^{8}}{8} - \frac{135 x ^{7}}{2} + 315 x^{6} - \frac{1701}{2} x^{5} + \frac{5103 x ^{4}}{4} - \frac{1701}{2} x^{3} + C

\int_{1}^{t} \frac{33}{x ^{3}} d x

\frac{d}{d x} (lo g_{6} (2 x^{2} + 4))

\int (\frac{10 x}{1 + 5 x ^{2}}) d x

d er i v a t i v e \frac{2 x ^{3}}{3}

\frac{d}{d x} (\frac{e ^{x}}{3 x ^{2}})