1. **Problem 1:** Show that $$\int_0^1 \frac{1}{\sqrt{x^2 + 3}} \, dx = \frac{1}{2} \ln 3$$. 2. **Step 1:** Use the substitution $$u = x + \sqrt{x^2 + 3}$$. 3. **Step 2:** Differentiate $$u$$ to find $$du$$ and express $$dx$$ in terms of $$du$$. 4. **Step 3:** Rewrite the integral in terms of $$u$$ and simplify. 5. **Step 4:** Evaluate the integral and apply the limits from 0 to 1. 6. **Result:** The integral evaluates to $$\frac{1}{2} \ln 3$$. 7. **Problem 2:** Let $$J = \int_0^1 \sqrt{x^2 + 3} \, dx$$. Show that $$2J = 2 + \int_0^1 \frac{3}{\sqrt{x^2 + 3}} \, dx$$. 8. **Step 1:** Use integration by parts with $$u = x$$ and $$dv = \sqrt{x^2 + 3} \, dx$$. 9. **Step 2:** Compute $$du = dx$$ and $$v$$ by integrating $$dv$$. 10. **Step 3:** Apply integration by parts formula: $$\int u \, dv = uv - \int v \, du$$. 11. **Step 4:** Simplify and rearrange to get the expression for $$2J$$. 12. **Step 5:** Use the result from Problem 1 to evaluate the integral and find $$J = 1 + \frac{3}{4} \ln 3$$. 13. **Problem 3:** Using the formula $$\int_0^a f(x) \, dx = \int_0^a f(a - x) \, dx$$, show that $$\int_0^{\pi/4} \ln \left( \frac{\cos x}{\cos x + \sin x} \right) dx = \frac{\pi}{8} \ln \frac{1}{2}$$. 14. **Step 1:** Let $$I = \int_0^{\pi/4} \ln \left( \frac{\cos x}{\cos x + \sin x} \right) dx$$. 15. **Step 2:** Use the substitution $$x \to \frac{\pi}{4} - x$$ and apply the given formula. 16. **Step 3:** Add the two expressions for $$I$$ and simplify using logarithm properties. 17. **Step 4:** Solve for $$I$$ to get $$I = \frac{\pi}{8} \ln \frac{1}{2}$$. **Final answers:** $$\int_0^1 \frac{1}{\sqrt{x^2 + 3}} \, dx = \frac{1}{2} \ln 3$$ $$J = \int_0^1 \sqrt{x^2 + 3} \, dx = 1 + \frac{3}{4} \ln 3$$ $$\int_0^{\pi/4} \ln \left( \frac{\cos x}{\cos x + \sin x} \right) dx = \frac{\pi}{8} \ln \frac{1}{2}$$