1. **State the problem:** We need to evaluate the integral $$\int (\cos(4\pi t) \mathbf{i} + \sin(3\pi t) \mathbf{j} + t^3 \mathbf{k}) \, dt$$ where $\mathbf{i}, \mathbf{j}, \mathbf{k}$ are unit vectors. 2. **Recall the integral rules:** The integral of a vector function is the vector of the integrals of its components. Also, recall: - $$\int \cos(ax) \, dx = \frac{\sin(ax)}{a} + C$$ - $$\int \sin(ax) \, dx = -\frac{\cos(ax)}{a} + C$$ - $$\int t^n \, dt = \frac{t^{n+1}}{n+1} + C$$ for $n \neq -1$. 3. **Integrate each component:** - For the $\mathbf{i}$ component: $$\int \cos(4\pi t) \, dt = \frac{\sin(4\pi t)}{4\pi} + C_1$$ - For the $\mathbf{j}$ component: $$\int \sin(3\pi t) \, dt = -\frac{\cos(3\pi t)}{3\pi} + C_2$$ - For the $\mathbf{k}$ component: $$\int t^3 \, dt = \frac{t^4}{4} + C_3$$ 4. **Combine the results:** $$\int (\cos(4\pi t) \mathbf{i} + \sin(3\pi t) \mathbf{j} + t^3 \mathbf{k}) \, dt = \frac{\sin(4\pi t)}{4\pi} \mathbf{i} - \frac{\cos(3\pi t)}{3\pi} \mathbf{j} + \frac{t^4}{4} \mathbf{k} + \mathbf{C}$$ where $\mathbf{C} = C_1 \mathbf{i} + C_2 \mathbf{j} + C_3 \mathbf{k}$ is the constant of integration vector. **Final answer:** $$\boxed{\frac{\sin(4\pi t)}{4\pi} \mathbf{i} - \frac{\cos(3\pi t)}{3\pi} \mathbf{j} + \frac{t^4}{4} \mathbf{k} + \mathbf{C}}$$