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

解答

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

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

求解步骤

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

将

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

用部份分式展开:

\frac{1}{3 ( x - 1 )} + \frac{5 x + 5}{3 ( x ^{2} + 2 )}

= \int \frac{1}{3 ( x - 1 )} + \frac{5 x + 5}{3 ( x ^{2} + 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 \frac{1}{3 ( x - 1 )} d x + \int \frac{5 x + 5}{3 ( x ^{2} + 2 )} d x

\int \frac{1}{3 ( x - 1 )} d x = \frac{1}{3} ln ∣ x - 1 ∣

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

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

解答补常数

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

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