1. **State the problem:** Evaluate the integral $$\int \cos 4\theta \sqrt{2} - \sin 4\theta \, d\theta$$. 2. **Rewrite the integral:** Separate the integral into two parts: $$\int \cos 4\theta \sqrt{2} \, d\theta - \int \sin 4\theta \, d\theta$$. 3. **Recall integration formulas:** - $$\int \cos(ax) \, dx = \frac{1}{a} \sin(ax) + C$$ - $$\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C$$ 4. **Apply the formulas:** - For $$\int \cos 4\theta \sqrt{2} \, d\theta$$, treat $$\sqrt{2}$$ as a constant multiplier: $$\sqrt{2} \int \cos 4\theta \, d\theta = \sqrt{2} \cdot \frac{1}{4} \sin 4\theta = \frac{\sqrt{2}}{4} \sin 4\theta$$. - For $$\int \sin 4\theta \, d\theta$$: $$-\frac{1}{4} \cos 4\theta$$. 5. **Combine results:** $$\int \cos 4\theta \sqrt{2} - \sin 4\theta \, d\theta = \frac{\sqrt{2}}{4} \sin 4\theta + \frac{1}{4} \cos 4\theta + C$$. 6. **Final answer:** $$\boxed{\frac{\sqrt{2}}{4} \sin 4\theta + \frac{1}{4} \cos 4\theta + C}$$. This integral was solved by splitting the integral, applying standard trigonometric integral formulas, and carefully handling constants.