integral of ((5x+3)/(x^2+2x-3))

解答

\int (\frac{5 x + 3}{x ^{2} + 2 x - 3}) d x

2 ln ∣ x - 1 ∣ + 3 ln ∣ x + 3 ∣ + C

求解步骤

\int \frac{5 x + 3}{x ^{2} + 2 x - 3} d x

将

\frac{5 x + 3}{x ^{2} + 2 x - 3}

用部份分式展开:

\frac{2}{x - 1} + \frac{3}{x + 3}

= \int \frac{2}{x - 1} + \frac{3}{x + 3} 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} d x + \int \frac{3}{x + 3} d x

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

\int \frac{3}{x + 3} d x = 3 ln ∣ x + 3 ∣

= 2 ln ∣ x - 1 ∣ + 3 ln ∣ x + 3 ∣

解答补常数

= 2 ln ∣ x - 1 ∣ + 3 ln ∣ x + 3 ∣ + C

\int \frac{( x + 8 ) ^{4}}{2 x} d x

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