integral from 1 to 2 of (x^2+8)/(3x-x^2)

解答

\int_{1}^{2} \frac{x ^{2} + 8}{3 x - x ^{2}} d x

- 1 + \frac{25}{3} ln (2)

十进制

4.77622 \dots

求解步骤

\int_{1}^{2} \frac{x ^{2} + 8}{3 x - x ^{2}} d x

分解

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

= \int_{1}^{2} \frac{x ^{2} + 8}{- x ( x - 3 )} d x

提出常数:

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

= - \int_{1}^{2} \frac{x ^{2} + 8}{x ( x - 3 )} d x

乘开

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

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_{1}^{2} \frac{x ^{2}}{x ( x - 3 )} d x + \int_{1}^{2} \frac{8}{x ( x - 3 )} d x)

\int_{1}^{2} \frac{x ^{2}}{x ( x - 3 )} d x = 1 - 3 ln (2)

\int_{1}^{2} \frac{8}{x ( x - 3 )} d x = - \frac{16}{3} ln (2)

= - (1 - 3 ln (2) - \frac{16}{3} ln (2))

化简

- (1 - 3 ln (2) - \frac{16}{3} ln (2)) : - 1 + \frac{25}{3} ln (2)

= - 1 + \frac{25}{3} ln (2)

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

d er i v a t i v e \frac{7 + 2 x}{5 - 8 x}

\frac{\partial}{\partial x} (\frac{1}{tan ( x y )})

\frac{d y}{d x} = (x + y + 2)^{2}

\frac{\partial}{\partial x} (e^{- 2 x} \cdot cos (2 y))