integral of 1/(x^3-512)

解答

\int \frac{1}{x ^{3} - 512} d x

\frac{1}{192} ln ∣ x - 8 ∣ - \frac{1}{384} ln x^{2} + 8 x + 64 - \frac{1}{64 3} arctan (\frac{1}{4 3} x + \frac{1}{3}) + C

求解步骤

\int \frac{1}{x ^{3} - 512} d x

将

\frac{1}{x ^{3} - 512}

用部份分式展开:

\frac{1}{192 ( x - 8 )} + \frac{- x - 16}{192 ( x ^{2} + 8 x + 64 )}

= \int \frac{1}{192 ( x - 8 )} + \frac{- x - 16}{192 ( x ^{2} + 8 x + 64 )} 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}{192 ( x - 8 )} d x + \int \frac{- x - 16}{192 ( x ^{2} + 8 x + 64 )} d x

\int \frac{1}{192 ( x - 8 )} d x = \frac{1}{192} ln ∣ x - 8 ∣

\int \frac{- x - 16}{192 ( x ^{2} + 8 x + 64 )} d x = \frac{1}{384 3} (- 3 ln x^{2} + 8 x + 64 - 6 arctan (\frac{1}{4 3} x + \frac{1}{3}))

= \frac{1}{192} ln ∣ x - 8 ∣ + \frac{1}{384 3} (- 3 ln x^{2} + 8 x + 64 - 6 arctan (\frac{1}{4 3} x + \frac{1}{3}))

乘开

\frac{1}{384 3} (- 3 ln x^{2} + 8 x + 64 - 6 arctan (\frac{1}{4 3} x + \frac{1}{3})) : - \frac{1}{384} ln x^{2} + 8 x + 64 - \frac{1}{64 3} arctan (\frac{1}{4 3} x + \frac{1}{3})

= \frac{1}{192} ln ∣ x - 8 ∣ - \frac{1}{384} ln x^{2} + 8 x + 64 - \frac{1}{64 3} arctan (\frac{1}{4 3} x + \frac{1}{3})

解答补常数

= \frac{1}{192} ln ∣ x - 8 ∣ - \frac{1}{384} ln x^{2} + 8 x + 64 - \frac{1}{64 3} arctan (\frac{1}{4 3} x + \frac{1}{3}) + C

(2 y - \frac{1}{x} + cos (3 x)) \frac{d y}{d x} + \frac{y}{x ^{2}} - 4 x^{3} - 3 y sin (3 x) = 0

\frac{d}{d x} ((cos (x^{2}))^{2})

\frac{d}{d x} (x^{x + 1})

s l o p e (- 20, - 4), (- 12, - 10)

n = 1 \sum \infty \frac{2}{25} \cdot (\frac{1}{5})^{n}