integral of ((x^2-1)^3)/(sqrt(x))

解答

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

\frac{2}{13} x^{\frac{13}{2}} - \frac{2}{3} x^{\frac{9}{2}} + \frac{6}{5} x^{\frac{5}{2}} - 2 x + C

求解步骤

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

乘开

\frac{( x ^{2} - 1 ) ^{3}}{x} : x^{\frac{11}{2}} - 3 x^{\frac{7}{2}} + 3 x^{\frac{3}{2}} - \frac{1}{x}

= \int x^{\frac{11}{2}} - 3 x^{\frac{7}{2}} + 3 x^{\frac{3}{2}} - \frac{1}{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^{\frac{11}{2}} d x - \int 3 x^{\frac{7}{2}} d x + \int 3 x^{\frac{3}{2}} d x - \int \frac{1}{x} d x

\int x^{\frac{11}{2}} d x = \frac{2}{13} x^{\frac{13}{2}}

\int 3 x^{\frac{7}{2}} d x = \frac{2}{3} x^{\frac{9}{2}}

\int 3 x^{\frac{3}{2}} d x = \frac{6}{5} x^{\frac{5}{2}}

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

= \frac{2}{13} x^{\frac{13}{2}} - \frac{2}{3} x^{\frac{9}{2}} + \frac{6}{5} x^{\frac{5}{2}} - 2 x

解答补常数

= \frac{2}{13} x^{\frac{13}{2}} - \frac{2}{3} x^{\frac{9}{2}} + \frac{6}{5} x^{\frac{5}{2}} - 2 x + C

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