Chapter 8 - Techniques of Integration - 8.7 Improper Integrals - Exercises - Page 440: 14

$$\frac{1}{2}$$

\begin{aligned} \int_{0}^{\infty} \frac{d x}{(x+1)^{3}} &=\lim _{R \rightarrow \infty} \int_{0}^{R} \frac{d x}{(x+1)^{3}} \\ &=\lim _{R \rightarrow \infty} \left.\frac{(x+1)^{-2}}{-2}\right|_{0} ^{R} \\ &=\lim _{R \rightarrow \infty} \frac{1}{2}-\frac{1}{2} \frac{1}{(R+1)^{2}} \\ &=\frac{1}{2} \end{aligned}