integral of sqrt(x)(x+1/x)

解答

\int x (x + \frac{1}{x}) d x

2 x + \frac{2}{5} x^{\frac{5}{2}} + C

求解步骤

\int x (x + \frac{1}{x}) d x

乘开

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

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

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

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

= 2 x + \frac{2}{5} x^{\frac{5}{2}}

解答补常数

= 2 x + \frac{2}{5} x^{\frac{5}{2}} + C