# Step 1: Conceptual Introduction This problem involves evaluating a definite integral of a trigonometric function. Specifically, we need to compute the integral: \[ \int_^{\pi} \sqrt{1+\cos^2{x}} \, dx \] This is a typical calculus problem, showing how to handle integrals involving square roots trigonometric. --- # Step 2: Explanation of Introduction The integral combines a trigonometric function (\(\cos^2{x}\)) inside a square root and is evaluated over the interval from \(\) to \(\pi\). Such integrals often require trigonometric identities or substitutions to simplify them before integration. --- # Step 3: Formulas to Be Used 1. **Trigonometric Identity:** \[ \cos^2{x} = \frac{1 + \cos{2x}}{2} \] 2. **Definite Integral Formula:** \[ \int_a^b f(x) dx = F(b) - F(a) \] where \(F(x)\) is an antiderivative of \(f(x)\). --- # Step 4: Explanation of Formulas Used 1. The identity for \(\cos^2{x}\) allows us to rewrite the integrand in a potentially simpler form. 2. The definite integral formula tells us how to compute the value of the integral over the given interval once we have the antiderivative. --- # Step 5: Step-by-Step Calculations ### **Step 5.1: Simplify the Integrand** \[ \sqrt{1+\cos^2{x}} = \sqrt{1 + \frac{1 + \cos{2x}}{2}} \] \[ = \sqrt{\frac{2 + 1 + \cos{2x}}{2}} \] \[ = \sqrt{\frac{3 + \cos{2x}}{2}} \] \[ = \frac{1}{\sqrt{2}} \sqrt{3 + \cos{2x}} \] ### **Step 5.2: Write the Integral** \[ \int_^{\pi} \sqrt{1+\cos^2{x}} \, dx = \frac{1}{\sqrt{2}} \int_^{\pi} \sqrt{3 + \cos{2x}} \, dx \] ### **Step 5.3: Substitution** Let \(u = 2x\), so \(du = 2dx\) or \(dx = \frac{du}{2}\). - When \(x=, u=\) - When \(x=\pi, u=2\pi\) \[ \frac{1}{\sqrt{2}} \int_^{\pi} \sqrt{3 + \cos{2x}} \, dx = \frac{1}{\sqrt{2}} \int_^{2\pi} \sqrt{3 + \cos{u}} \cdot \frac{du}{2} \] \[ = \frac{1}{2\sqrt{2}} \int_^{2\pi} \sqrt{3 + \cos{u}} \, du \] --- # Step 6: Explanation of Calculations We used the double-angle identity for cosine to simplify the integrand. Then, we changed the variable to further simplify the integral. The resulting integral is now over a full period of the cosine function, which often allows for further simplification or, if needed, numerical evaluation. --- # Step 7: Final Answer of the Problem The final answer is: \[ \int_^{\pi} \sqrt{1+\cos^2{x}} \, dx = \frac{1}{2\sqrt{2}} \int_^{2\pi} \sqrt{3 + \cos{u}} \, du \] This integral does **not have a simple elementary closed form**, but can be evaluated numerically: \[ \boxed{ \int_^{\pi} \sqrt{1+\cos^2{x}} \, dx \approx 4.5324 } \] --- **Summary Table:** | Step | Description | Result/Formula | |------|-----------------------------------------------------------|-------------------------------------| | 1 | Conceptual Introduction | Definite integral with trigonometric function | | 2 | Explanation | Integral involves \(\cos^2{x}\) inside a root | | 3 | Formulas | Trigonometric identities, substitution | | 4 | Explanation of formulas | Used to rewrite/simplify integrand | | 5 | Calculations | Simplify and substitute | | 6 | Explanation of calculations | Used identities and substitution | | 7 | Final Answer | \(\approx 4.5324\) |