1. **State the problem:** Integrate the function $$\frac{x^3-2}{x^2+1}$$ using long division. 2. **Perform long division:** Divide the numerator $$x^3-2$$ by the denominator $$x^2+1$$. 3. **Divide:** $$x^3 \div x^2 = x$$, so the first term of the quotient is $$x$$. 4. **Multiply:** Multiply the divisor by the quotient term: $$x \times (x^2+1) = x^3 + x$$. 5. **Subtract:** $$ (x^3 - 2) - (x^3 + x) = -x - 2$$. 6. **Express the original function:** $$\frac{x^3-2}{x^2+1} = x + \frac{-x - 2}{x^2 + 1}$$. 7. **Rewrite:** $$x - \frac{x + 2}{x^2 + 1}$$. 8. **Set up the integral:** $$\int \frac{x^3-2}{x^2+1} dx = \int x dx - \int \frac{x + 2}{x^2 + 1} dx$$. 9. **Integrate the first term:** $$\int x dx = \frac{x^2}{2} + C_1$$. 10. **Split the second integral:** $$\int \frac{x + 2}{x^2 + 1} dx = \int \frac{x}{x^2 + 1} dx + \int \frac{2}{x^2 + 1} dx$$. 11. **For the first part:** Let $$u = x^2 + 1$$, so $$du = 2x dx$$, then $$\int \frac{x}{x^2+1} dx = \frac{1}{2} \int \frac{du}{u} = \frac{1}{2} \ln|x^2 +1| + C_2$$. 12. **For the second part:** $$\int \frac{2}{x^2 +1} dx = 2 \arctan x + C_3$$. 13. **Combine results:** $$\int \frac{x^3-2}{x^2+1} dx = \frac{x^2}{2} - \left( \frac{1}{2} \ln|x^2 +1| + 2 \arctan x \right) + C$$ where $$C$$ combines all constants. 14. **Final answer:** $$\boxed{ \int \frac{x^3-2}{x^2+1} dx = \frac{x^2}{2} - \frac{1}{2} \ln|x^2 +1| - 2 \arctan x + C }$$