integral of (x^2-3x+5)/(sqrt(x))

解答

\int \frac{x ^{2} - 3 x + 5}{x} d x

\frac{2}{5} x^{\frac{5}{2}} - 2 x^{\frac{3}{2}} + 10 x + C

求解步骤

\int \frac{x ^{2} - 3 x + 5}{x} d x

乘开

\frac{x ^{2} - 3 x + 5}{x} : x^{\frac{3}{2}} - 3 x + \frac{5}{x}

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

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

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

\int \frac{5}{x} d x = 10 x

= \frac{2}{5} x^{\frac{5}{2}} - 2 x^{\frac{3}{2}} + 10 x

解答补常数

= \frac{2}{5} x^{\frac{5}{2}} - 2 x^{\frac{3}{2}} + 10 x + C

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