Step 1: Write down the function to be integrated. We need to integrate $y = 4x^3 + 2x + \cos x$. This means we need to find $\int (4x^3 + 2x + \cos x) \,dx$. Step 2: Apply the sum rule for integration, integrating each term separately. $$\int (4x^3 + 2x + \cos x) \,dx = \int 4x^3 \,dx + \int 2x \,dx + \int \cos x \,dx$$ Step 3: Integrate each term using the power rule $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$ and the standard integral for $\cos x$. For the first term: $$\int 4x^3 \,dx = 4 \cdot \frac{x^{3+1}}{3+1} = 4 \cdot \frac{x^4}{4} = x^4$$ For the second term: $$\int 2x \,dx = 2 \cdot \frac{x^{1+1}}{1+1} = 2 \cdot \frac{x^2}{2} = x^2$$ For the third term: $$\int \cos x \,dx = \sin x$$ Step 4: Combine the integrated terms and add the constant of integration, $C$. $$x^4 + x^2 + \sin x + C$$ Step 5: Compare the result with the given options. The calculated integral is $x^4 + x^2 + \sin x + C$, which matches option (C). The final answer is $\boxed{\text{C}}$. Last free one today — make it count tomorrow, or type /upgrade for unlimited.