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

\frac{d}{d x} (- \frac{cos ( 2 x )}{2})

\int t^{5} ln (t) d t

e^{(e^{(x)} - x)}

f (x) = tan (4 x^{2} - 3 x - 4)

f (x) = 2 x^{2} + 3 x, (- 3, 9)

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