integral of (2x-4+8sqrt(x))/(sqrt(x))

解答

\int \frac{2 x - 4 + 8 x}{x} d x

\frac{4}{3} x^{\frac{3}{2}} - 8 x + 8 x + C

求解步骤

\int \frac{2 x - 4 + 8 x}{x} d x

乘开

\frac{2 x - 4 + 8 x}{x} : 2 x - \frac{4}{x} + 8

= \int 2 x - \frac{4}{x} + 8 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 2 x d x - \int \frac{4}{x} d x + \int 8 d x

\int 2 x d x = \frac{4}{3} x^{\frac{3}{2}}

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

\int 8 d x = 8 x

= \frac{4}{3} x^{\frac{3}{2}} - 8 x + 8 x

解答补常数

= \frac{4}{3} x^{\frac{3}{2}} - 8 x + 8 x + C

x \to 1 lim (\frac{x ^{2} - 3 x}{x})

y^{^{'}} + \frac{4}{5} y = 1 - \frac{1}{4} t

\int \frac{2}{1 - t ^{2}} d t

\int \frac{e ^{3 x}}{x} d x

\frac{\partial}{\partial x} (- 6 e^{y})