What is the integral of (-x^3-x^2-4x+28)/(x^3+4x) ?

The integral of (-x^3-x^2-4x+28)/(x^3+4x) is -x-4ln|x^2+4|+7ln|x|+C

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integral of (-x^3-x^2-4x+28)/(x^3+4x)

\int \frac{- x ^{3} - x ^{2} - 4 x + 28}{x ^{3} + 4 x} d x

- x - 4 ln x^{2} + 4 + 7 ln ∣ x ∣ + C

Solution steps

\int \frac{- x ^{3} - x ^{2} - 4 x + 28}{x ^{3} + 4 x} d x

Expand

\frac{- x ^{3} - x ^{2} - 4 x + 28}{x ^{3} + 4 x} : - \frac{x ^{3}}{x ^{3} + 4 x} - \frac{x ^{2}}{x ^{3} + 4 x} - \frac{4 x}{x ^{3} + 4 x} + \frac{28}{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{x ^{3}}{x ^{3} + 4 x} d x - \int \frac{x ^{2}}{x ^{3} + 4 x} d x - \int \frac{4 x}{x ^{3} + 4 x} d x + \int \frac{28}{x ^{3} + 4 x} d x

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

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

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

\int \frac{28}{x ^{3} + 4 x} d x = 28 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4)

= - (- 2 arctan (\frac{x}{2}) + x) - \frac{1}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2}) + 28 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4)

Simplify

- (- 2 arctan (\frac{x}{2}) + x) - \frac{1}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2}) + 28 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4) : - x - 4 ln x^{2} + 4 + 7 ln ∣ x ∣

= - x - 4 ln x^{2} + 4 + 7 ln ∣ x ∣

Add a constant to the solution

= - x - 4 ln x^{2} + 4 + 7 ln ∣ x ∣ + C

\int_{0}^{7} \frac{8}{9 x ^{2} + 10 x + 1} d x

\frac{\partial}{\partial x} (e^{3 y} cos (- 3 x))

y = e^{x}, y = 9 - e^{x}, x = - 1, x = 1

(sin (ln (t)))^{2}

What is the integral of (-x^3-x^2-4x+28)/(x^3+4x) ?
The integral of (-x^3-x^2-4x+28)/(x^3+4x) is -x-4ln|x^2+4|+7ln|x|+C