integral of (3x-2)/(x^2+3x-10)

解答

\int \frac{3 x - 2}{x ^{2} + 3 x - 10} d x

\frac{4}{7} ln ∣ x - 2 ∣ + \frac{17}{7} ln ∣ x + 5 ∣ + C

求解步骤

\int \frac{3 x - 2}{x ^{2} + 3 x - 10} d x

将

\frac{3 x - 2}{x ^{2} + 3 x - 10}

用部份分式展开:

\frac{4}{7 ( x - 2 )} + \frac{17}{7 ( x + 5 )}

= \int \frac{4}{7 ( x - 2 )} + \frac{17}{7 ( x + 5 )} 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{4}{7 ( x - 2 )} d x + \int \frac{17}{7 ( x + 5 )} d x

\int \frac{4}{7 ( x - 2 )} d x = \frac{4}{7} ln ∣ x - 2 ∣

\int \frac{17}{7 ( x + 5 )} d x = \frac{17}{7} ln ∣ x + 5 ∣

= \frac{4}{7} ln ∣ x - 2 ∣ + \frac{17}{7} ln ∣ x + 5 ∣

解答补常数

= \frac{4}{7} ln ∣ x - 2 ∣ + \frac{17}{7} ln ∣ x + 5 ∣ + C

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