Chapter 7 - Principles Of Integral Evaluation - 7.8 Improper Integrals - Exercises Set 7.8 - Page 554: 15

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$$\eqalign{ & \int_{ - \infty }^\infty {\frac{x}{{{{\left( {{x^2} + 3} \right)}^2}}}} dx \cr & {\text{Use the definition }}\int_{ - \infty }^\infty {f\left( x \right)} dx = \int_{ - \infty }^c {f\left( x \right)} dx + \int_c^{ + \infty } {f\left( x \right)} dx \cr & {\text{Let }}c = 0. \cr & \int_{ - \infty }^\infty {\frac{x}{{{{\left( {{x^2} + 3} \right)}^2}}}} dx = \int_{ - \infty }^0 {\frac{x}{{{{\left( {{x^2} + 3} \right)}^2}}}} dx + \int_0^{ + \infty } {\frac{x}{{{{\left( {{x^2} + 3} \right)}^2}}}} dx \cr & {\text{Evaluating the integrals on the right side separately}} \cr & \int_{ - \infty }^0 {\frac{x}{{{{\left( {{x^2} + 3} \right)}^2}}}} dx = \mathop {\lim }\limits_{a \to - \infty } \int_a^0 {\frac{x}{{{{\left( {{x^2} + 3} \right)}^2}}}} dx \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\lim }\limits_{a \to - \infty } \frac{1}{2}\int_a^0 {{{\left( {{x^2} + 3} \right)}^{ - 2}}\left( {2x} \right)} dx \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{2}\mathop {\lim }\limits_{a \to - \infty } \left[ {\frac{{{{\left( {{x^2} + 3} \right)}^{ - 1}}}}{{ - 1}}} \right]_a^0 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \frac{1}{2}\mathop {\lim }\limits_{a \to - \infty } \left[ {\frac{1}{{{x^2} + 3}}} \right]_a^0 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \frac{1}{2}\mathop {\lim }\limits_{a \to - \infty } \left[ {\frac{1}{{{{\left( 0 \right)}^2} + 3}} - \frac{1}{{{{\left( a \right)}^2} + 3}}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \frac{1}{2}\mathop {\lim }\limits_{a \to - \infty } \left[ {\frac{1}{3} - \frac{1}{{{a^2} + 3}}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Find the limit when }}a \to - \infty \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \frac{1}{2}\left[ {\frac{1}{3} - \frac{1}{{{{\left( { - \infty } \right)}^2} + 3}}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \frac{1}{2}\left[ {\frac{1}{3} - 0} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \frac{1}{6} \cr & and \cr & \int_0^{ + \infty } {\frac{x}{{{{\left( {{x^2} + 3} \right)}^2}}}} dx = \mathop {\lim }\limits_{b \to + \infty } \int_0^b {\frac{x}{{{{\left( {{x^2} + 3} \right)}^2}}}} dx \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\lim }\limits_{b \to + \infty } \frac{1}{2}\int_0^b {{{\left( {{x^2} + 3} \right)}^{ - 2}}\left( {2x} \right)} dx \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{2}\mathop {\lim }\limits_{b \to + \infty } \left[ {\frac{{{{\left( {{x^2} + 3} \right)}^{ - 1}}}}{{ - 1}}} \right]_0^b \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \frac{1}{2}\mathop {\lim }\limits_{b \to + \infty } \left[ {\frac{1}{{{x^2} + 3}}} \right]_0^b \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \frac{1}{2}\mathop {\lim }\limits_{b \to + \infty } \left[ {\frac{1}{{{{\left( b \right)}^2} + 3}} - \frac{1}{{{{\left( 0 \right)}^2} + 3}}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \frac{1}{2}\mathop {\lim }\limits_{b \to + \infty } \left[ {\frac{1}{{{b^2} + 3}} - \frac{1}{3}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Find the limit when }}b \to + \infty \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \frac{1}{2}\left[ {\frac{1}{{{{\left( { + \infty } \right)}^2} + 3}} - \frac{1}{3}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \frac{1}{2}\left[ {0 - \frac{1}{3}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{6} \cr & {\text{Thus}}{\text{,}} \cr & \int_{ - \infty }^\infty {\frac{x}{{{{\left( {{x^2} + 3} \right)}^2}}}} dx = \int_{ - \infty }^0 {\frac{x}{{{{\left( {{x^2} + 3} \right)}^2}}}} dx + \int_0^{ + \infty } {\frac{x}{{{{\left( {{x^2} + 3} \right)}^2}}}} dx \cr & \int_{ - \infty }^\infty {\frac{x}{{{{\left( {{x^2} + 3} \right)}^2}}}} dx = - \frac{1}{6} + \frac{1}{6} \cr & \int_{ - \infty }^\infty {\frac{x}{{{{\left( {{x^2} + 3} \right)}^2}}}} dx = 0 \cr} $$