1. **Problem:** Evaluate the integral $$\int \frac{2e^{2}\sinh x}{e^{2-x} + e^{2+x}} \, dx$$. 2. **Recall the definitions and formulas:** - Hyperbolic sine: $$\sinh x = \frac{e^x - e^{-x}}{2}$$. - Simplify the denominator: $$e^{2-x} + e^{2+x} = e^2 e^{-x} + e^2 e^x = e^2 (e^{-x} + e^x)$$. 3. **Rewrite the integral:** $$\int \frac{2 e^{2} \sinh x}{e^{2} (e^{-x} + e^{x})} \, dx = \int \frac{2 \sinh x}{e^{-x} + e^{x}} \, dx$$. 4. **Substitute $$\sinh x$$ and simplify:** $$\frac{2 \sinh x}{e^{-x} + e^{x}} = \frac{2 \cdot \frac{e^{x} - e^{-x}}{2}}{e^{-x} + e^{x}} = \frac{e^{x} - e^{-x}}{e^{-x} + e^{x}}$$. 5. **Recognize the expression:** $$\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} = \tanh x$$. 6. **Integral becomes:** $$\int \tanh x \, dx$$. 7. **Recall integral of $$\tanh x$$:** $$\int \tanh x \, dx = \ln |\cosh x| + C$$. 8. **Final answer:** $$\boxed{\ln |\cosh x| + C}$$.