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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

f (x) = \frac{1}{x ^{4}}, at x = 3

\frac{\partial}{\partial z} (x e^{z})

\int x^{2} + 3 x - 1000000 d x

\frac{1}{x + 9}

\int \frac{e ^{\frac{x}{2}}}{x} d x

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