integral of (x^2-5x-7)/((x^2+9)(x-2))

解答

\int \frac{x ^{2} - 5 x - 7}{( x ^{2} + 9 ) ( x - 2 )} d x

ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3}) - ln ∣ x - 2 ∣ + C

求解步骤

\int \frac{x ^{2} - 5 x - 7}{( x ^{2} + 9 ) ( x - 2 )} d x

将

\frac{x ^{2} - 5 x - 7}{( x ^{2} + 9 ) ( x - 2 )}

用部份分式展开:

\frac{2 x - 1}{x ^{2} + 9} - \frac{1}{x - 2}

= \int \frac{2 x - 1}{x ^{2} + 9} - \frac{1}{x - 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{2 x - 1}{x ^{2} + 9} d x - \int \frac{1}{x - 2} d x

\int \frac{2 x - 1}{x ^{2} + 9} d x = ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3})

\int \frac{1}{x - 2} d x = ln ∣ x - 2 ∣

= ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3}) - ln ∣ x - 2 ∣

解答补常数

= ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3}) - ln ∣ x - 2 ∣ + C

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