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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

Add a constant to the solution

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

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