integral of (5x^2+1)/(x^3+1)

解答

\int \frac{5 x ^{2} + 1}{x ^{3} + 1} d x

\frac{5}{3} ln x^{3} + 1 + \frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3})) + C

求解步骤

\int \frac{5 x ^{2} + 1}{x ^{3} + 1} d x

乘开

\frac{5 x ^{2} + 1}{x ^{3} + 1} : \frac{5 x ^{2}}{x ^{3} + 1} + \frac{1}{x ^{3} + 1}

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{5 x ^{2}}{x ^{3} + 1} d x + \int \frac{1}{x ^{3} + 1} d x

\int \frac{5 x ^{2}}{x ^{3} + 1} d x = \frac{5}{3} ln x^{3} + 1

\int \frac{1}{x ^{3} + 1} d x = \frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3}))

= \frac{5}{3} ln x^{3} + 1 + \frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3}))

解答补常数

= \frac{5}{3} ln x^{3} + 1 + \frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3})) + C

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