What is the integral of (x^4-x^2)/(x^2+1) ?

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

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integral of (x^4-x^2)/(x^2+1)

\int \frac{x ^{4} - x ^{2}}{x ^{2} + 1} d x

\frac{x ^{3}}{3} - 2 x + 2 arctan (x) + C

Solution steps

\int \frac{x ^{4} - x ^{2}}{x ^{2} + 1} d x

Expand

\frac{x ^{4} - x ^{2}}{x ^{2} + 1} : \frac{x ^{4}}{x ^{2} + 1} - \frac{x ^{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 ^{4}}{x ^{2} + 1} d x - \int \frac{x ^{2}}{x ^{2} + 1} d x

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

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

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

Simplify

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

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

Add a constant to the solution

= \frac{x ^{3}}{3} - 2 x + 2 arctan (x) + C

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