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

解答

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

x + \frac{7}{6} ln ∣ x + 1 ∣ - \frac{9}{2} ln ∣ x - 1 ∣ + \frac{16}{3} ln ∣ x - 2 ∣ + C

求解步骤

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

将

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

用部份分式展开:

1 + \frac{7}{6 ( x + 1 )} - \frac{9}{2 ( x - 1 )} + \frac{16}{3 ( x - 2 )}

= \int 1 + \frac{7}{6 ( x + 1 )} - \frac{9}{2 ( x - 1 )} + \frac{16}{3 ( 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 1 d x + \int \frac{7}{6 ( x + 1 )} d x - \int \frac{9}{2 ( x - 1 )} d x + \int \frac{16}{3 ( x - 2 )} d x

\int 1 d x = x

\int \frac{7}{6 ( x + 1 )} d x = \frac{7}{6} ln ∣ x + 1 ∣

\int \frac{9}{2 ( x - 1 )} d x = \frac{9}{2} ln ∣ x - 1 ∣

\int \frac{16}{3 ( x - 2 )} d x = \frac{16}{3} ln ∣ x - 2 ∣

= x + \frac{7}{6} ln ∣ x + 1 ∣ - \frac{9}{2} ln ∣ x - 1 ∣ + \frac{16}{3} ln ∣ x - 2 ∣

解答补常数

= x + \frac{7}{6} ln ∣ x + 1 ∣ - \frac{9}{2} ln ∣ x - 1 ∣ + \frac{16}{3} ln ∣ x - 2 ∣ + C

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

a re a y = 2 x, x = 3

\int e^{- s t} e^{a t} d t

\int 2 x^{4} + 5 x + 1 d x

\frac{y ^{^{'}}}{y} = 5