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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

\int 1 d x = x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{2}{x ^{2}} d x = - \frac{2}{x}

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

= x + ln ∣ x ∣ - (- \frac{2}{x}) - 5 ln ∣ x + 2 ∣

Simplify

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

Add a constant to the solution

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

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