What is the integral of (3x^3+x^2+5x+1)/(x^3+x) ?

The integral of (3x^3+x^2+5x+1)/(x^3+x) is (3x^3)/(x^2+1)+2arctan(x)+(3x)/(x^2+1)+ln|x|+C

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integral of (3x^3+x^2+5x+1)/(x^3+x)

\int \frac{3 x ^{3} + x ^{2} + 5 x + 1}{x ^{3} + x} d x

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

Solution steps

\int \frac{3 x ^{3} + x ^{2} + 5 x + 1}{x ^{3} + x} d x

Expand

\frac{3 x ^{3} + x ^{2} + 5 x + 1}{x ^{3} + x} : \frac{3 x ^{3}}{x ^{3} + x} + \frac{x ^{2}}{x ^{3} + x} + \frac{5 x}{x ^{3} + x} + \frac{1}{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{x ^{2}}{x ^{3} + x} d x + \int \frac{5 x}{x ^{3} + x} d x + \int \frac{1}{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{x ^{2}}{x ^{3} + x} d x = \frac{1}{2} ln x^{2} + 1

\int \frac{5 x}{x ^{3} + x} d x = 5 arctan (x)

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

= 3 (\frac{x ^{3}}{x ^{2} + 1} - arctan (x) + \frac{x}{1 + x ^{2}}) + \frac{1}{2} ln x^{2} + 1 + 5 arctan (x) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1

Simplify

3 (\frac{x ^{3}}{x ^{2} + 1} - arctan (x) + \frac{x}{1 + x ^{2}}) + \frac{1}{2} ln x^{2} + 1 + 5 arctan (x) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1 : \frac{3 x ^{3}}{x ^{2} + 1} + 2 arctan (x) + \frac{3 x}{x ^{2} + 1} + ln ∣ x ∣

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

Add a constant to the solution

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

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What is the integral of (3x^3+x^2+5x+1)/(x^3+x) ?
The integral of (3x^3+x^2+5x+1)/(x^3+x) is (3x^3)/(x^2+1)+2arctan(x)+(3x)/(x^2+1)+ln|x|+C