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

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

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

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

\frac{x ^{3}}{3} - 2 x + 22 arctan (\frac{x}{2}) + C

Solution steps

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

Long division

\frac{x ^{4}}{x ^{2} + 2} : x^{2} - 2 + \frac{4}{x ^{2} + 2}

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

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

\int 2 d x = 2 x

\int \frac{4}{x ^{2} + 2} d x = 22 arctan (\frac{x}{2})

= \frac{x ^{3}}{3} - 2 x + 22 arctan (\frac{x}{2})

Add a constant to the solution

= \frac{x ^{3}}{3} - 2 x + 22 arctan (\frac{x}{2}) + C

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