integral of 1/((x-2)(x-4)(x+1))

解答

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

- \frac{1}{6} ln ∣ x - 2 ∣ + \frac{1}{10} ln ∣ x - 4 ∣ + \frac{1}{15} ln ∣ x + 1 ∣ + C

求解步骤

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

将

\frac{1}{( x - 2 ) ( x - 4 ) ( x + 1 )}

用部份分式展开:

- \frac{1}{6 ( x - 2 )} + \frac{1}{10 ( x - 4 )} + \frac{1}{15 ( x + 1 )}

= \int - \frac{1}{6 ( x - 2 )} + \frac{1}{10 ( x - 4 )} + \frac{1}{15 ( x + 1 )} 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}{6 ( x - 2 )} d x + \int \frac{1}{10 ( x - 4 )} d x + \int \frac{1}{15 ( x + 1 )} d x

\int \frac{1}{6 ( x - 2 )} d x = \frac{1}{6} ln ∣ x - 2 ∣

\int \frac{1}{10 ( x - 4 )} d x = \frac{1}{10} ln ∣ x - 4 ∣

\int \frac{1}{15 ( x + 1 )} d x = \frac{1}{15} ln ∣ x + 1 ∣

= - \frac{1}{6} ln ∣ x - 2 ∣ + \frac{1}{10} ln ∣ x - 4 ∣ + \frac{1}{15} ln ∣ x + 1 ∣

解答补常数

= - \frac{1}{6} ln ∣ x - 2 ∣ + \frac{1}{10} ln ∣ x - 4 ∣ + \frac{1}{15} ln ∣ x + 1 ∣ + C

d er i v a t i v e x + 2

a re a x = - y^{2} + 4 y, [1, 4]

x \to 0 lim (sin^{2} (\frac{π}{x}))

\frac{\partial}{\partial x} ((9 x + 2 y)^{10})

f (x) = e^{2 x} + 1