1. The problem is to find the integral of the function $1 + u^2$ with respect to $u$. 2. Recall that the integral of a sum is the sum of the integrals, so we can write: $$\int (1 + u^2) \, du = \int 1 \, du + \int u^2 \, du$$ 3. The integral of 1 with respect to $u$ is $u$ because the derivative of $u$ is 1. 4. The integral of $u^2$ with respect to $u$ is given by the power rule for integration: $$\int u^n \, du = \frac{u^{n+1}}{n+1} + C$$ Applying this for $n=2$: $$\int u^2 \, du = \frac{u^{3}}{3} + C$$ 5. Combining these results, the integral is: $$u + \frac{u^{3}}{3} + C$$ 6. Therefore, the integral of $1 + u^2$ with respect to $u$ is: $$\boxed{u + \frac{u^{3}}{3} + C}$$