Integral Substitution 652Df2

1. **State the problem:** We need to evaluate the integral $$\int x(x^2 + 2)^3 \, dx.$$ 2. **Choose a substitution:** Let $$u = x^2 + 2$$ because the expression inside the parentheses is raised to a power and its derivative appears outside. 3. **Find differential $$du$$:** Differentiate $$u$$ with respect to $$x$$: $$du = 2x \, dx \implies x \, dx = \frac{du}{2}.$$ 4. **Rewrite the integral in terms of $$u$$:** Substitute $$u$$ and $$x \, dx$$ into the integral: $$\int x (x^2 + 2)^3 \, dx = \int (u)^3 \cdot \frac{du}{2} = \frac{1}{2} \int u^3 \, du.$$ 5. **Integrate with respect to $$u$$:** Use the power rule for integration: $$\int u^3 \, du = \frac{u^4}{4} + C.$$ 6. **Substitute back to $$x$$:** Replace $$u$$ with $$x^2 + 2$$: $$\frac{1}{2} \cdot \frac{(x^2 + 2)^4}{4} + C = \frac{(x^2 + 2)^4}{8} + C.$$ **Final answer:** $$\int x(x^2 + 2)^3 \, dx = \frac{(x^2 + 2)^4}{8} + C.$$