integral of (5x^2-2)/(x^2-2x-24)

解答

\int \frac{5 x ^{2} - 2}{x ^{2} - 2 x - 24} d x

5 x - \frac{39}{5} ln ∣ x + 4 ∣ + \frac{89}{5} ln ∣ x - 6 ∣ + C

求解步骤

\int \frac{5 x ^{2} - 2}{x ^{2} - 2 x - 24} d x

将

\frac{5 x ^{2} - 2}{x ^{2} - 2 x - 24}

用部份分式展开:

5 - \frac{39}{5 ( x + 4 )} + \frac{89}{5 ( x - 6 )}

= \int 5 - \frac{39}{5 ( x + 4 )} + \frac{89}{5 ( x - 6 )} 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 5 d x - \int \frac{39}{5 ( x + 4 )} d x + \int \frac{89}{5 ( x - 6 )} d x

\int 5 d x = 5 x

\int \frac{39}{5 ( x + 4 )} d x = \frac{39}{5} ln ∣ x + 4 ∣

\int \frac{89}{5 ( x - 6 )} d x = \frac{89}{5} ln ∣ x - 6 ∣

= 5 x - \frac{39}{5} ln ∣ x + 4 ∣ + \frac{89}{5} ln ∣ x - 6 ∣

解答补常数

= 5 x - \frac{39}{5} ln ∣ x + 4 ∣ + \frac{89}{5} ln ∣ x - 6 ∣ + C

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