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

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

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

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

\frac{4 x ^{3}}{x ^{2} + 1} + \frac{4 x}{x ^{2} + 1} + 4 ln ∣ x ∣ - 2 ln x^{2} + 1 + C

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

= \frac{4 x ^{3}}{x ^{2} + 1} + \frac{4 x}{x ^{2} + 1} + 4 ln ∣ x ∣ - 2 ln x^{2} + 1

Add a constant to the solution

= \frac{4 x ^{3}}{x ^{2} + 1} + \frac{4 x}{x ^{2} + 1} + 4 ln ∣ x ∣ - 2 ln x^{2} + 1 + C

4 (x - \frac{1}{x})^{2}, at x = 2

f (x) = 4 x^{7} - 5 cos (x)

\int e^{x} x d x

n = 1 \sum \infty \frac{n !}{1 8 ^{n}}

ln (sin (11 x))

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