integral of ((3x+1)/(sqrt(x)))

解答

\int (\frac{3 x + 1}{x}) d x

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

求解步骤

\int \frac{3 x + 1}{x} d x

乘开

\frac{3 x + 1}{x} : 3 x + \frac{1}{x}

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

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

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

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

解答补常数

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

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