integral of 1/((x-3)(x+2))

解答

\int \frac{1}{( x - 3 ) ( x + 2 )} d x

\frac{1}{5} ln ∣ x - 3 ∣ - \frac{1}{5} ln ∣ x + 2 ∣ + C

求解步骤

\int \frac{1}{( x - 3 ) ( x + 2 )} d x

将

\frac{1}{( x - 3 ) ( x + 2 )}

用部份分式展开:

\frac{1}{5 ( x - 3 )} - \frac{1}{5 ( x + 2 )}

= \int \frac{1}{5 ( x - 3 )} - \frac{1}{5 ( x + 2 )} 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 \frac{1}{5 ( x - 3 )} d x - \int \frac{1}{5 ( x + 2 )} d x

\int \frac{1}{5 ( x - 3 )} d x = \frac{1}{5} ln ∣ x - 3 ∣

\int \frac{1}{5 ( x + 2 )} d x = \frac{1}{5} ln ∣ x + 2 ∣

= \frac{1}{5} ln ∣ x - 3 ∣ - \frac{1}{5} ln ∣ x + 2 ∣

解答补常数

= \frac{1}{5} ln ∣ x - 3 ∣ - \frac{1}{5} ln ∣ x + 2 ∣ + C

s l o p e (1, 6), (0.5, 8)

x \to 0 lim ((1 - 2 x)^{\frac{1}{2}})

in v erse l a pl a ce \frac{150}{s ( s ^{2} + 8 s + 25 )}

\frac{\partial}{\partial x} (\frac{1}{f ( x )})

x \to 1 - lim (\frac{1}{1 - x})