integral of (x^2+5x-8)/(sqrt(x))

解答

\int \frac{x ^{2} + 5 x - 8}{x} d x

\frac{2}{5} x^{\frac{5}{2}} + \frac{10}{3} x^{\frac{3}{2}} - 16 x + C

求解步骤

\int \frac{x ^{2} + 5 x - 8}{x} d x

乘开

\frac{x ^{2} + 5 x - 8}{x} : x^{\frac{3}{2}} + 5 x - \frac{8}{x}

= \int x^{\frac{3}{2}} + 5 x - \frac{8}{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 x^{\frac{3}{2}} d x + \int 5 x d x - \int \frac{8}{x} d x

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

\int 5 x d x = \frac{10}{3} x^{\frac{3}{2}}

\int \frac{8}{x} d x = 16 x

= \frac{2}{5} x^{\frac{5}{2}} + \frac{10}{3} x^{\frac{3}{2}} - 16 x

解答补常数

= \frac{2}{5} x^{\frac{5}{2}} + \frac{10}{3} x^{\frac{3}{2}} - 16 x + C

x \to \infty lim (2 x + 8)

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

x \to 2 lim (a^{2} x + 2 a)

\int \frac{1}{( 1 + 4 y ^{2} )} d y

\frac{d}{d x} (e^{- \frac{1}{2} x})