1. **State the problem:** Find the derivative $f^\prime(x)$ of the function $f(x) = x^4 + 2x$. 2. **Formula and rules:** The derivative of a function $f(x)$ is found using the power rule: if $f(x) = x^n$, then $f^\prime(x) = nx^{n-1}$. 3. **Calculate the derivative:** $$f^\prime(x) = \frac{d}{dx}(x^4) + \frac{d}{dx}(2x) = 4x^3 + 2$$ 4. **State the problem:** Find the integral $$\int \frac{2x^3 + 1}{x^4 + 2x} \, dx$$ 5. **Rewrite the integral:** Notice that the denominator is $f(x) = x^4 + 2x$ and the numerator is $2x^3 + 1$. 6. **Check if numerator relates to derivative:** We have $f^\prime(x) = 4x^3 + 2$, which is twice the numerator: $$2x^3 + 1 = \frac{1}{2} (4x^3 + 2) = \frac{1}{2} f^\prime(x)$$ 7. **Use substitution:** Let $u = x^4 + 2x$, then $du = f^\prime(x) dx = (4x^3 + 2) dx$. 8. **Rewrite integral in terms of $u$:** $$\int \frac{2x^3 + 1}{x^4 + 2x} dx = \int \frac{\frac{1}{2} f^\prime(x)}{u} dx = \frac{1}{2} \int \frac{du}{u}$$ 9. **Integrate:** $$\frac{1}{2} \int \frac{du}{u} = \frac{1}{2} \ln|u| + C = \frac{1}{2} \ln|x^4 + 2x| + C$$ **Final answers:** $$f^\prime(x) = 4x^3 + 2$$ $$\int \frac{2x^3 + 1}{x^4 + 2x} dx = \frac{1}{2} \ln|x^4 + 2x| + C$$