1. **Problem Statement:** Calculate the integral $$\int \frac{x^2}{(x+1)^3} \, dx$$ which is a typical integral involving rational functions and useful in economic modeling. 2. **Formula and Rules:** We use substitution and integration by parts for such integrals. Important rule: simplify the integrand if possible and use substitution to reduce complexity. 3. **Step 1: Substitution** Let $$u = x+1 \Rightarrow x = u-1$$ and $$dx = du$$. 4. **Step 2: Rewrite the integral** $$\int \frac{x^2}{(x+1)^3} \, dx = \int \frac{(u-1)^2}{u^3} \, du = \int \frac{u^2 - 2u + 1}{u^3} \, du = \int \left(\frac{u^2}{u^3} - \frac{2u}{u^3} + \frac{1}{u^3}\right) du = \int \left(u^{-1} - 2u^{-2} + u^{-3}\right) du$$ 5. **Step 3: Integrate term-by-term** $$\int u^{-1} du = \ln|u| + C$$ $$\int u^{-2} du = \int u^{-2} du = -u^{-1} + C$$ $$\int u^{-3} du = -\frac{1}{2} u^{-2} + C$$ 6. **Step 4: Combine results** $$\int \left(u^{-1} - 2u^{-2} + u^{-3}\right) du = \ln|u| - 2(-u^{-1}) - \frac{1}{2} u^{-2} + C = \ln|u| + 2u^{-1} - \frac{1}{2} u^{-2} + C$$ 7. **Step 5: Substitute back to x** $$= \ln|x+1| + \frac{2}{x+1} - \frac{1}{2(x+1)^2} + C$$ **Final answer:** $$\int \frac{x^2}{(x+1)^3} \, dx = \ln|x+1| + \frac{2}{x+1} - \frac{1}{2(x+1)^2} + C$$