1. **Problem statement:** Evaluate the integral $$\int 2x(x^2 + 4)^3 \, dx$$ using the substitution $$u = x^2 + 4$$. 2. **Formula and substitution:** We use substitution for integrals where a part of the integrand is the derivative of another part. Here, let $$u = x^2 + 4$$. 3. **Find $$du$$:** Differentiate $$u$$ with respect to $$x$$: $$du = 2x \, dx$$ This matches the $$2x \, dx$$ part in the integral. 4. **Rewrite the integral:** Substitute $$u$$ and $$du$$ into the integral: $$\int 2x(x^2 + 4)^3 \, dx = \int u^3 \, du$$ 5. **Integrate:** Use the power rule for integration: $$\int u^3 \, du = \frac{u^{4}}{4} + C$$ 6. **Back-substitute:** Replace $$u$$ with $$x^2 + 4$$: $$\frac{(x^2 + 4)^4}{4} + C$$ **Final answer:** $$\int 2x(x^2 + 4)^3 \, dx = \frac{(x^2 + 4)^4}{4} + C$$