integral of (2x^2-4x+3)/((x^2+2)(x+1))

解答

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

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

求解步骤

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

将

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

用部份分式展开:

\frac{- x - 3}{x ^{2} + 2} + \frac{3}{x + 1}

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

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

\int \frac{3}{x + 1} d x = 3 ln ∣ x + 1 ∣

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

解答补常数

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

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