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

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

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

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

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

Solution steps

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

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

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

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

Add a constant to the solution

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

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