integral of x/(5-5x^3)

解答

\int \frac{x}{5 - 5 x ^{3}} d x

- \frac{1}{15} ln ∣ x - 1 ∣ + \frac{1}{30} (ln 4 x^{2} + 4 x + 4 - 23 arctan (\frac{2 x + 1}{3})) + C

求解步骤

\int \frac{x}{5 - 5 x ^{3}} d x

将

\frac{x}{5 - 5 x ^{3}}

用部份分式展开:

- \frac{1}{15 ( x - 1 )} + \frac{x - 1}{15 ( x ^{2} + x + 1 )}

= \int - \frac{1}{15 ( x - 1 )} + \frac{x - 1}{15 ( x ^{2} + 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}{15 ( x - 1 )} d x + \int \frac{x - 1}{15 ( x ^{2} + x + 1 )} d x

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

\int \frac{x - 1}{15 ( x ^{2} + x + 1 )} d x = \frac{1}{30} (ln 4 x^{2} + 4 x + 4 - 23 arctan (\frac{2 x + 1}{3}))

= - \frac{1}{15} ln ∣ x - 1 ∣ + \frac{1}{30} (ln 4 x^{2} + 4 x + 4 - 23 arctan (\frac{2 x + 1}{3}))

解答补常数

= - \frac{1}{15} ln ∣ x - 1 ∣ + \frac{1}{30} (ln 4 x^{2} + 4 x + 4 - 23 arctan (\frac{2 x + 1}{3})) + C

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