1. **State the problem:** We want to find the integral $$\int \sqrt{1 + u^2} \, du$$. 2. **Recall the formula:** The integral of $$\sqrt{1 + u^2}$$ with respect to $$u$$ is given by the formula: $$\int \sqrt{1 + u^2} \, du = \frac{u}{2} \sqrt{1 + u^2} + \frac{1}{2} \ln \left| u + \sqrt{1 + u^2} \right| + C$$ 3. **Explanation of the formula:** - The term $$\frac{u}{2} \sqrt{1 + u^2}$$ comes from using integration by parts or a suitable substitution. - The logarithmic term $$\frac{1}{2} \ln \left| u + \sqrt{1 + u^2} \right|$$ arises from integrating the remaining part after substitution. - $$C$$ is the constant of integration, representing any constant value since indefinite integrals are determined up to an additive constant. 4. **Step-by-step derivation (optional):** - Use substitution or integration by parts to handle the integral. - For example, set $$u = \sinh t$$, then $$\sqrt{1 + u^2} = \cosh t$$, and the integral becomes simpler. - After integrating and back-substituting, you arrive at the formula above. 5. **Summary:** The integral of $$\sqrt{1 + u^2}$$ is a combination of an algebraic term and a logarithmic term as shown, which is a standard result in integral calculus. This formula is useful in problems involving arc lengths and hyperbolic functions.