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

解答

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

\frac{1}{3} (\frac{2}{3} x^{\frac{3}{2}} + 4 x + 8 x) + C

求解步骤

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

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

乘开

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

= \frac{1}{3} \cdot \int x + 4 + \frac{4}{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

= \frac{1}{3} (\int x d x + \int 4 d x + \int \frac{4}{x} d x)

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

\int 4 d x = 4 x

\int \frac{4}{x} d x = 8 x

= \frac{1}{3} (\frac{2}{3} x^{\frac{3}{2}} + 4 x + 8 x)

解答补常数

= \frac{1}{3} (\frac{2}{3} x^{\frac{3}{2}} + 4 x + 8 x) + C

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