integral of (3x^2+8x-12)/(x^3+7x^2+12x)

解答

\int \frac{3 x ^{2} + 8 x - 12}{x ^{3} + 7 x ^{2} + 12 x} d x

- ln ∣ x ∣ + 3 ln ∣ x + 3 ∣ + ln ∣ x + 4 ∣ + C

求解步骤

\int \frac{3 x ^{2} + 8 x - 12}{x ^{3} + 7 x ^{2} + 12 x} d x

将

\frac{3 x ^{2} + 8 x - 12}{x ^{3} + 7 x ^{2} + 12 x}

用部份分式展开:

- \frac{1}{x} + \frac{3}{x + 3} + \frac{1}{x + 4}

= \int - \frac{1}{x} + \frac{3}{x + 3} + \frac{1}{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 \frac{1}{x} d x + \int \frac{3}{x + 3} d x + \int \frac{1}{x + 4} d x

\int \frac{1}{x} d x = ln ∣ x ∣

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

\int \frac{1}{x + 4} d x = ln ∣ x + 4 ∣

= - ln ∣ x ∣ + 3 ln ∣ x + 3 ∣ + ln ∣ x + 4 ∣

解答补常数

= - ln ∣ x ∣ + 3 ln ∣ x + 3 ∣ + ln ∣ x + 4 ∣ + C

\int x^{3} x^{4} + 8 d x

\int x e^{\frac{x ^{2}}{5}} d x

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