1. **State the problem:** We need to find the integral of the function $$\frac{x^2}{x^2 + 1}$$ with respect to $x$. 2. **Rewrite the integrand:** Notice that $$\frac{x^2}{x^2 + 1} = \frac{x^2 + 1 - 1}{x^2 + 1} = 1 - \frac{1}{x^2 + 1}$$. 3. **Split the integral:** Using linearity of integration, $$\int \frac{x^2}{x^2 + 1} dx = \int 1 \, dx - \int \frac{1}{x^2 + 1} dx$$. 4. **Integrate each term:** - $$\int 1 \, dx = x + C_1$$ - $$\int \frac{1}{x^2 + 1} dx = \arctan(x) + C_2$$ 5. **Combine results:** $$\int \frac{x^2}{x^2 + 1} dx = x - \arctan(x) + C$$ where $C = C_1 - C_2$ is the constant of integration. **Final answer:** $$\boxed{\int \frac{x^2}{x^2 + 1} dx = x - \arctan(x) + C}$$