# Chapter 7 - Principles Of Integral Evaluation - 7.3 Integrating Trigonometric Functions - Exercises Set 7.3 - Page 507: 25

$$\ln \left| {\cos {e^{ - x}}} \right| + C$$

#### Work Step by Step

\eqalign{ & \int {{e^{ - x}}\tan \left( {{e^{ - x}}} \right)} dx \cr & = \int {\frac{{\sin {e^{ - x}}}}{{\cos {e^{ - x}}}}} \left( {{e^{ - x}}} \right)dx \cr & {\text{substitute }}u = \cos {e^{ - x}},{\text{ }}du = {e^{ - x}}\sin {e^{ - x}}dx \cr & = \int {\frac{{du}}{u}} \cr & {\text{find the antiderivative }} \cr & = \ln \left| u \right| + C \cr & {\text{write in terms of }}x,{\text{ replace }}u = \cos {e^{ - x}} \cr & \ln \left| {\cos {e^{ - x}}} \right| + C \cr}

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