What is the integral of (x^2+12x+12)/(x^3-4x) ?

The integral of (x^2+12x+12)/(x^3-4x) is -3ln|x|-ln|x+2|+5ln|x-2|+C

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integral of (x^2+12x+12)/(x^3-4x)

\int \frac{x ^{2} + 12 x + 12}{x ^{3} - 4 x} d x

- 3 ln ∣ x ∣ - ln ∣ x + 2 ∣ + 5 ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{x ^{2} + 12 x + 12}{x ^{3} - 4 x} d x

Take the partial fraction of

\frac{x ^{2} + 12 x + 12}{x ^{3} - 4 x} : - \frac{3}{x} - \frac{1}{x + 2} + \frac{5}{x - 2}

= \int - \frac{3}{x} - \frac{1}{x + 2} + \frac{5}{x - 2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{3}{x} d x - \int \frac{1}{x + 2} d x + \int \frac{5}{x - 2} d x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

\int \frac{5}{x - 2} d x = 5 ln ∣ x - 2 ∣

= - 3 ln ∣ x ∣ - ln ∣ x + 2 ∣ + 5 ln ∣ x - 2 ∣

Add a constant to the solution

= - 3 ln ∣ x ∣ - ln ∣ x + 2 ∣ + 5 ln ∣ x - 2 ∣ + C

\int x^{2} csc^{2} (x) cot (x) d x

\int + sin^{3} (x) cos^{14} (x) d x

\frac{3 x}{x - \frac{3}{x}}

\frac{35}{4 + x + x}

\frac{d}{d x} (\frac{x - 6}{x + 4})

What is the integral of (x^2+12x+12)/(x^3-4x) ?
The integral of (x^2+12x+12)/(x^3-4x) is -3ln|x|-ln|x+2|+5ln|x-2|+C