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

The integral of ((x^3-4))/((x^3+2x^2+2x)) is x-2ln|x|+2arctan(x+1)+C

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

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

x - 2 ln ∣ x ∣ + 2 arctan (x + 1) + C

Solution steps

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

Expand

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

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

\int \frac{4}{x ^{3} + 2 x ^{2} + 2 x} d x = 2 ln ∣ x ∣ - ln x^{2} + 2 x + 2 - 2 arctan (x + 1)

= - ln x^{2} + 2 x + 2 + x - (2 ln ∣ x ∣ - ln x^{2} + 2 x + 2 - 2 arctan (x + 1))

Simplify

= x - 2 ln ∣ x ∣ + 2 arctan (x + 1)

Add a constant to the solution

= x - 2 ln ∣ x ∣ + 2 arctan (x + 1) + C

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What is the integral of ((x^3-4))/((x^3+2x^2+2x)) ?
The integral of ((x^3-4))/((x^3+2x^2+2x)) is x-2ln|x|+2arctan(x+1)+C