What is the integral of (2x^3+x^2+8x-12)/(x^3+4x) ?

The integral of (2x^3+x^2+8x-12)/(x^3+4x) is 2x+2ln|x^2+4|-3ln|x|+C

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integral of (2x^3+x^2+8x-12)/(x^3+4x)

\int \frac{2 x ^{3} + x ^{2} + 8 x - 12}{x ^{3} + 4 x} d x

2 x + 2 ln x^{2} + 4 - 3 ln ∣ x ∣ + C

Solution steps

\int \frac{2 x ^{3} + x ^{2} + 8 x - 12}{x ^{3} + 4 x} d x

Expand

\frac{2 x ^{3} + x ^{2} + 8 x - 12}{x ^{3} + 4 x} : \frac{2 x ^{3}}{x ^{3} + 4 x} + \frac{x ^{2}}{x ^{3} + 4 x} + \frac{8 x}{x ^{3} + 4 x} - \frac{12}{x ^{3} + 4 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{2 x ^{3}}{x ^{3} + 4 x} d x + \int \frac{x ^{2}}{x ^{3} + 4 x} d x + \int \frac{8 x}{x ^{3} + 4 x} d x - \int \frac{12}{x ^{3} + 4 x} d x

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

\int \frac{x ^{2}}{x ^{3} + 4 x} d x = \frac{1}{2} ln x^{2} + 4

\int \frac{8 x}{x ^{3} + 4 x} d x = 4 arctan (\frac{x}{2})

\int \frac{12}{x ^{3} + 4 x} d x = 12 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4)

= 4 (- arctan (\frac{x}{2}) + \frac{x}{2}) + \frac{1}{2} ln x^{2} + 4 + 4 arctan (\frac{x}{2}) - 12 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4)

Simplify

4 (- arctan (\frac{x}{2}) + \frac{x}{2}) + \frac{1}{2} ln x^{2} + 4 + 4 arctan (\frac{x}{2}) - 12 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4) : 2 x + 2 ln x^{2} + 4 - 3 ln ∣ x ∣

= 2 x + 2 ln x^{2} + 4 - 3 ln ∣ x ∣

Add a constant to the solution

= 2 x + 2 ln x^{2} + 4 - 3 ln ∣ x ∣ + C

(0) (1.2)

35 t^{2}

\frac{d y}{d x} = \frac{4 ( 1 - y ) ^{2}}{( 1 - 2 x ) ^{2}}

\frac{d}{d x} (1 + \frac{1}{x ^{2}})

\frac{d}{d x} (- 6 sin (2 x))

What is the integral of (2x^3+x^2+8x-12)/(x^3+4x) ?
The integral of (2x^3+x^2+8x-12)/(x^3+4x) is 2x+2ln|x^2+4|-3ln|x|+C