integral of (x+2)/((x-1)(x^2+4))

解答

\int \frac{x + 2}{( x - 1 ) ( x ^{2} + 4 )} d x

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

求解步骤

\int \frac{x + 2}{( x - 1 ) ( x ^{2} + 4 )} d x

将

\frac{x + 2}{( x - 1 ) ( x ^{2} + 4 )}

用部份分式展开:

\frac{3}{5 ( x - 1 )} + \frac{- 3 x + 2}{5 ( x ^{2} + 4 )}

= \int \frac{3}{5 ( x - 1 )} + \frac{- 3 x + 2}{5 ( x ^{2} + 4 )} 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{3}{5 ( x - 1 )} d x + \int \frac{- 3 x + 2}{5 ( x ^{2} + 4 )} d x

\int \frac{3}{5 ( x - 1 )} d x = \frac{3}{5} ln ∣ x - 1 ∣

\int \frac{- 3 x + 2}{5 ( x ^{2} + 4 )} d x = \frac{1}{5} (- \frac{3}{2} ln x^{2} + 4 + arctan (\frac{x}{2}))

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

解答补常数

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

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