Chapter 8 - Further Techniques and Applications of Integration - 8.4 Improper Integrals - 8.4 Exercises - Page 452: 19

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$$\eqalign{ & \int_0^\infty {\frac{{dx}}{{{{\left( {x + 1} \right)}^2}}}} \cr & {\text{by the definition of an improper integral}}{\text{}} \cr & \int_0^\infty {\frac{{dx}}{{{{\left( {x + 1} \right)}^2}}}} = \mathop {\lim }\limits_{b \to \infty } \int_0^b {\frac{1}{{{{\left( {x + 1} \right)}^2}}}} dx \cr & {\text{write as}} \cr & = \mathop {\lim }\limits_{b \to \infty } \int_0^b {{{\left( {x + 1} \right)}^{ - 2}}} dx \cr & {\text{integrating by using the power rule}} \cr & = \mathop {\lim }\limits_{b \to \infty } \left[ {\frac{{{{\left( {x + 1} \right)}^{ - 2 + 1}}}}{{ - 2 + 1}}} \right]_0^b \cr & = \mathop {\lim }\limits_{b \to \infty } \left[ {\frac{{{{\left( {x + 1} \right)}^{ - 1}}}}{{ - 1}}} \right]_0^b \cr & = - \mathop {\lim }\limits_{b \to \infty } \left[ {\frac{1}{{x + 1}}} \right]_0^b \cr & or \cr & = \mathop {\lim }\limits_{b \to \infty } \left[ {\frac{1}{{x + 1}}} \right]_b^0 \cr & {\text{ using ftc}} \cr & = \mathop {\lim }\limits_{b \to \infty } \left[ {\frac{1}{{0 + 1}} - \frac{1}{{b + 1}}} \right] \cr & = \mathop {\lim }\limits_{b \to \infty } \left[ {1 - \frac{1}{{b + 1}}} \right] \cr & {\text{evaluating the limit when }}b \to \infty \cr & = 1 - \frac{1}{{\infty + 1}} \cr & = 1 - 0 \cr & = 1 \cr} $$