integral of (5x^2-6)/(x^2-2x-8)

解答

\int \frac{5 x ^{2} - 6}{x ^{2} - 2 x - 8} d x

5 x - \frac{7}{3} ln ∣ x + 2 ∣ + \frac{37}{3} ln ∣ x - 4 ∣ + C

求解步骤

\int \frac{5 x ^{2} - 6}{x ^{2} - 2 x - 8} d x

将

\frac{5 x ^{2} - 6}{x ^{2} - 2 x - 8}

用部份分式展开:

5 - \frac{7}{3 ( x + 2 )} + \frac{37}{3 ( x - 4 )}

= \int 5 - \frac{7}{3 ( x + 2 )} + \frac{37}{3 ( x - 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 5 d x - \int \frac{7}{3 ( x + 2 )} d x + \int \frac{37}{3 ( x - 4 )} d x

\int 5 d x = 5 x

\int \frac{7}{3 ( x + 2 )} d x = \frac{7}{3} ln ∣ x + 2 ∣

\int \frac{37}{3 ( x - 4 )} d x = \frac{37}{3} ln ∣ x - 4 ∣

= 5 x - \frac{7}{3} ln ∣ x + 2 ∣ + \frac{37}{3} ln ∣ x - 4 ∣

解答补常数

= 5 x - \frac{7}{3} ln ∣ x + 2 ∣ + \frac{37}{3} ln ∣ x - 4 ∣ + C