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

解答

\int \frac{( x + 2 ) ^{2}}{5 x} d x

\frac{1}{5} (\frac{2}{3} x^{\frac{3}{2}} + 4 x + 8 x) + C

求解步骤

\int \frac{( x + 2 ) ^{2}}{5 x} d x

提出常数:

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

= \frac{1}{5} \cdot \int \frac{( x + 2 ) ^{2}}{x} d x

乘开

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

= \frac{1}{5} \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}{5} (\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}{5} (\frac{2}{3} x^{\frac{3}{2}} + 4 x + 8 x)

解答补常数

= \frac{1}{5} (\frac{2}{3} x^{\frac{3}{2}} + 4 x + 8 x) + C

\frac{d}{d x} (\frac{9 x}{ln ( x )})

f (x) = \frac{x + 1}{x}

t an g e n t (x - \frac{6}{x})^{8}

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

d er i v a t i v e 2 sec (x) + 5 cot (x)