What is the integral of (x^2+132x+132)/(x^3-4x) ?

The integral of (x^2+132x+132)/(x^3-4x) is -33ln|x|-16ln|x+2|+50ln|x-2|+C

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integral of (x^2+132x+132)/(x^3-4x)

\int \frac{x ^{2} + 132 x + 132}{x ^{3} - 4 x} d x

- 33 ln ∣ x ∣ - 16 ln ∣ x + 2 ∣ + 50 ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{x ^{2} + 132 x + 132}{x ^{3} - 4 x} d x

Take the partial fraction of

\frac{x ^{2} + 132 x + 132}{x ^{3} - 4 x} : - \frac{33}{x} - \frac{16}{x + 2} + \frac{50}{x - 2}

= \int - \frac{33}{x} - \frac{16}{x + 2} + \frac{50}{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{33}{x} d x - \int \frac{16}{x + 2} d x + \int \frac{50}{x - 2} d x

\int \frac{33}{x} d x = 33 ln ∣ x ∣

\int \frac{16}{x + 2} d x = 16 ln ∣ x + 2 ∣

\int \frac{50}{x - 2} d x = 50 ln ∣ x - 2 ∣

= - 33 ln ∣ x ∣ - 16 ln ∣ x + 2 ∣ + 50 ln ∣ x - 2 ∣

Add a constant to the solution

= - 33 ln ∣ x ∣ - 16 ln ∣ x + 2 ∣ + 50 ln ∣ x - 2 ∣ + C

What is the integral of (x^2+132x+132)/(x^3-4x) ?
The integral of (x^2+132x+132)/(x^3-4x) is -33ln|x|-16ln|x+2|+50ln|x-2|+C