integral of (x^2-8x-17)/((x-1)^2(x^2+1))

解答

\int \frac{x ^{2} - 8 x - 17}{( x - 1 ) ^{2} ( x ^{2} + 1 )} d x

9 ln ∣ x - 1 ∣ + \frac{12}{x - 1} - \frac{9}{2} ln x^{2} + 1 + 4 arctan (x) + C

求解步骤

\int \frac{x ^{2} - 8 x - 17}{( x - 1 ) ^{2} ( x ^{2} + 1 )} d x

将

\frac{x ^{2} - 8 x - 17}{( x - 1 ) ^{2} ( x ^{2} + 1 )}

用部份分式展开:

\frac{9}{x - 1} - \frac{12}{( x - 1 ) ^{2}} + \frac{- 9 x + 4}{x ^{2} + 1}

= \int \frac{9}{x - 1} - \frac{12}{( x - 1 ) ^{2}} + \frac{- 9 x + 4}{x ^{2} + 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{9}{x - 1} d x - \int \frac{12}{( x - 1 ) ^{2}} d x + \int \frac{- 9 x + 4}{x ^{2} + 1} d x

\int \frac{9}{x - 1} d x = 9 ln ∣ x - 1 ∣

\int \frac{12}{( x - 1 ) ^{2}} d x = - \frac{12}{x - 1}

\int \frac{- 9 x + 4}{x ^{2} + 1} d x = - \frac{9}{2} ln x^{2} + 1 + 4 arctan (x)

= 9 ln ∣ x - 1 ∣ - (- \frac{12}{x - 1}) - \frac{9}{2} ln x^{2} + 1 + 4 arctan (x)

化简

= 9 ln ∣ x - 1 ∣ + \frac{12}{x - 1} - \frac{9}{2} ln x^{2} + 1 + 4 arctan (x)

解答补常数

= 9 ln ∣ x - 1 ∣ + \frac{12}{x - 1} - \frac{9}{2} ln x^{2} + 1 + 4 arctan (x) + C

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