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

The integral of (x^3-2x)/(x^3+x) is x-3arctan(x)+C

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

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

x - 3 arctan (x) + C

Solution steps

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

Simplify

\frac{x ^{3} - 2 x}{x ^{3} + x} : \frac{x ^{2} - 2}{x ^{2} + 1}

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

Expand

\frac{x ^{2} - 2}{x ^{2} + 1} : \frac{x ^{2}}{x ^{2} + 1} - \frac{2}{x ^{2} + 1}

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 ^{2}}{x ^{2} + 1} d x - \int \frac{2}{x ^{2} + 1} d x

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

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

= - arctan (x) + x - 2 arctan (x)

Simplify

= x - 3 arctan (x)

Add a constant to the solution

= x - 3 arctan (x) + C

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