1. **State the problem:** We need to evaluate the integral $$\int e^{2x} \cos x \left(e^{2x}\right) \, dx$$. 2. **Simplify the integrand:** Notice that the integrand is $$e^{2x} \cos x \left(e^{2x}\right) = e^{2x} \cos x \cdot e^{2x} = e^{4x} \cos x$$. 3. **Rewrite the integral:** So the integral becomes $$\int e^{4x} \cos x \, dx$$. 4. **Use integration by parts or formula:** The integral of the form $$\int e^{ax} \cos(bx) \, dx$$ is given by $$\int e^{ax} \cos(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \cos(bx) + b \sin(bx)) + C$$ where $a=4$ and $b=1$. 5. **Apply the formula:** $$\int e^{4x} \cos x \, dx = \frac{e^{4x}}{4^2 + 1^2} (4 \cos x + 1 \sin x) + C = \frac{e^{4x}}{17} (4 \cos x + \sin x) + C$$. 6. **Final answer:** $$\boxed{\int e^{2x} \cos x \left(e^{2x}\right) \, dx = \frac{e^{4x}}{17} (4 \cos x + \sin x) + C}$$