integral of sqrt(x)(x^3-2/(x^2)+6)

解答

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

\frac{4}{x} + \frac{2}{9} x^{\frac{9}{2}} + 4 x^{\frac{3}{2}} + C

求解步骤

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

乘开

x (x^{3} - \frac{2}{x ^{2}} + 6) : - \frac{2}{x ^{\frac{3}{2}}} + x^{\frac{7}{2}} + 6 x^{\frac{1}{2}}

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

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

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

\int 6 x^{\frac{1}{2}} d x = 4 x^{\frac{3}{2}}

= - (- \frac{4}{x}) + \frac{2}{9} x^{\frac{9}{2}} + 4 x^{\frac{3}{2}}

化简

= \frac{4}{x} + \frac{2}{9} x^{\frac{9}{2}} + 4 x^{\frac{3}{2}}

解答补常数

= \frac{4}{x} + \frac{2}{9} x^{\frac{9}{2}} + 4 x^{\frac{3}{2}} + C

x \to 0 lim (x^{2} - \frac{cos ( x )}{10000})

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