integral of (2-3x^2)/(sqrt(5x^3-10x+5))

解答

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

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

求解步骤

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

使用换元积分法

= \int - \frac{1}{5 u} d u

提出常数:

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

= - \frac{1}{5} \cdot \int \frac{1}{u} d u

使用幂法则

= - \frac{1}{5} \cdot 2 u^{\frac{1}{2}}

u = 5 x^{3} - 10 x + 5

代回

= - \frac{1}{5} \cdot 2 (5 x^{3} - 10 x + 5)^{\frac{1}{2}}

化简

- \frac{1}{5} \cdot 2 (5 x^{3} - 10 x + 5)^{\frac{1}{2}} : - \frac{2}{5} 5 x^{3} - 10 x + 5

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

解答补常数

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

\frac{\partial}{\partial x} (x^{2} y e^{\frac{y}{z}})

\frac{\partial}{\partial x} (x + y)

x \to \infty lim (4 x^{3} (e^{- \frac{2}{x ^{3}}} - 1))

\int sin^{4} (d) x d x

a re a x = y^{2} + 6, y = \frac{x}{2} - \frac{3}{2}