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

The integral of (x^4)/(x^2-2) is (x^3)/3+2x-sqrt(2)ln| x/(sqrt(2))+1|+sqrt(2)ln| x/(sqrt(2))-1|+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 - 2 ln \frac{x}{2} + 1 + 2 ln \frac{x}{2} - 1 + 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 = - 2 ln \frac{x}{2} + 1 + 2 ln \frac{x}{2} - 1

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

Add a constant to the solution

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

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