Chapter 7 - Principles Of Integral Evaluation - 7.8 Improper Integrals - Exercises Set 7.8 - Page 554: 16

$$\frac{\pi }{2}$$

$$\eqalign{ & \int_{ - \infty }^\infty {\frac{{{e^{ - t}}}}{{1 + {e^{ - 2t}}}}} dt \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{{{e^{ - t}}}}{{1 + {e^{ - 2t}}}}} dx = \int_{ - \infty }^0 {\frac{{{e^{ - t}}}}{{1 + {e^{ - 2t}}}}} dx + \int_0^{ + \infty } {\frac{{{e^{ - t}}}}{{1 + {e^{ - 2t}}}}} dx \cr & {\text{Evaluating the integrals on the right side separately}} \cr & \int_{ - \infty }^0 {\frac{{{e^{ - t}}}}{{1 + {e^{ - 2t}}}}} dx = \mathop {\lim }\limits_{a \to - \infty } \int_a^0 {\frac{{{e^{ - t}}}}{{1 + {e^{ - 2t}}}}} dx \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \mathop {\lim }\limits_{a \to - \infty } \frac{1}{2}\int_a^0 {\frac{{ - {e^{ - t}}}}{{1 + {{\left( {{e^{ - t}}} \right)}^2}}}} dx \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \mathop {\lim }\limits_{a \to - \infty } \left[ {{{\tan }^{ - 1}}\left( {{e^{ - t}}} \right)} \right]_a^0 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \mathop {\lim }\limits_{a \to - \infty } \left[ {{{\tan }^{ - 1}}\left( {{e^{ - 0}}} \right) - {{\tan }^{ - 1}}\left( {{e^{ - a}}} \right)} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \mathop {\lim }\limits_{a \to - \infty } \left[ {\frac{\pi }{4} - {{\tan }^{ - 1}}\left( {{e^{ - a}}} \right)} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Find the limit when }}a \to - \infty \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \left[ {\frac{\pi }{4} - {{\tan }^{ - 1}}\left( \infty \right)} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \left[ {\frac{\pi }{4} - \frac{\pi }{2}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{\pi }{4} \cr & and \cr & \int_0^{ + \infty } {\frac{{{e^{ - t}}}}{{1 + {e^{ - 2t}}}}} dx = \mathop {\lim }\limits_{b \to + \infty } \int_0^b {\frac{{{e^{ - t}}}}{{1 + {e^{ - 2t}}}}} dx \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \mathop {\lim }\limits_{b \to + \infty } \int_0^b {\frac{{ - {e^{ - t}}}}{{1 + {{\left( {{e^{ - t}}} \right)}^2}}}} dx \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \mathop {\lim }\limits_{b \to + \infty } \left[ {{{\tan }^{ - 1}}\left( {{e^{ - t}}} \right)} \right]_0^b \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \mathop {\lim }\limits_{b \to + \infty } \left[ {{{\tan }^{ - 1}}\left( {{e^{ - b}}} \right) - {{\tan }^{ - 1}}\left( {{e^{ - 0}}} \right)} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \mathop {\lim }\limits_{b \to + \infty } \left[ {{{\tan }^{ - 1}}\left( {{e^{ - b}}} \right) - {{\tan }^{ - 1}}\left( 1 \right)} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \mathop {\lim }\limits_{b \to + \infty } \left[ {{{\tan }^{ - 1}}\left( {{e^{ - b}}} \right) - \frac{\pi }{4}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Find the limit when }}b \to + \infty \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \left[ {{{\tan }^{ - 1}}\left( {{e^{ - \infty }}} \right) - \frac{\pi }{4}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \left[ {0 - \frac{\pi }{4}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{\pi }{4} \cr & {\text{Thus}}{\text{,}} \cr & \int_{ - \infty }^\infty {\frac{{{e^{ - t}}}}{{1 + {e^{ - 2t}}}}} dx = \int_{ - \infty }^0 {\frac{{{e^{ - t}}}}{{1 + {e^{ - 2t}}}}} dx + \int_0^{ + \infty } {\frac{{{e^{ - t}}}}{{1 + {e^{ - 2t}}}}} dx \cr & \int_{ - \infty }^\infty {\frac{{{e^{ - t}}}}{{1 + {e^{ - 2t}}}}} dx = \frac{\pi }{4} + \frac{\pi }{4} \cr & \int_{ - \infty }^\infty {\frac{{{e^{ - t}}}}{{1 + {e^{ - 2t}}}}} dx = \frac{\pi }{2} \cr} $$