\[\begin{array}{l}
\int\limits_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {\frac{{dx}}{{{{\cos }^2}x\left( {1 + {e^{ - 3x}}} \right)}}} = \int\limits_{\frac{\pi }{4}}^{ - \frac{\pi }{4}} {\frac{{ - dt}}{{{{\cos }^2}\left( { - t} \right)\left( {1 + {e^{3t}}} \right)}}} = \int\limits_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {\frac{{dt}}{{{{\cos }^2}t\left( {1 + {e^{3t}}} \right)}}} = \int\limits_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {\frac{{dx}}{{{{\cos }^2}x\left( {1 + {e^{3x}}} \right)}}} \\
\int\limits_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {\frac{{dx}}{{{{\cos }^2}x\left( {1 + {e^{ - 3x}}} \right)}}} = \int\limits_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {\frac{{{e^{3x}}dx}}{{{{\cos }^2}x\left( {1 + {e^{3x}}} \right)}}} \\
= > 2\int\limits_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {\frac{{dx}}{{{{\cos }^2}x\left( {1 + {e^{ - 3x}}} \right)}}} = \int\limits_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {\frac{{dx}}{{{{\cos }^2}x\left( {1 + {e^{3x}}} \right)}}} + \int\limits_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {\frac{{{e^{3x}}dx}}{{{{\cos }^2}x\left( {1 + {e^{3x}}} \right)}}} = \int\limits_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {\frac{{dx}}{{{{\cos }^2}x}} = } t\left. {anx} \right|_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} = 2
\end{array}\]
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