integral of (x^2+2)(sqrt(x)-2)

解答

\int (x^{2} + 2) (x - 2) d x

\frac{2}{7} x^{\frac{7}{2}} - \frac{2 x ^{3}}{3} + \frac{4}{3} x^{\frac{3}{2}} - 4 x + C

求解步骤

\int (x^{2} + 2) (x - 2) d x

乘开

(x^{2} + 2) (x - 2) : x^{\frac{5}{2}} - 2 x^{2} + 2 x - 4

= \int x^{\frac{5}{2}} - 2 x^{2} + 2 x - 4 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{5}{2}} d x - \int 2 x^{2} d x + \int 2 x d x - \int 4 d x

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

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

\int 2 x d x = \frac{4}{3} x^{\frac{3}{2}}

\int 4 d x = 4 x

= \frac{2}{7} x^{\frac{7}{2}} - \frac{2 x ^{3}}{3} + \frac{4}{3} x^{\frac{3}{2}} - 4 x

解答补常数

= \frac{2}{7} x^{\frac{7}{2}} - \frac{2 x ^{3}}{3} + \frac{4}{3} x^{\frac{3}{2}} - 4 x + C

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