Chapter 6 - Exponential, Logarithmic, And Inverse Trigonometric Functions - 6.3 Derivatives Of Inverse Functions; Derivatives And Integrals Involving Exponential Functions - Exercises Set 6.3 - Page 433: 71

$$ - {e^{ - x}} + C$$

$$\eqalign{ & \int {\frac{{dx}}{{{e^x}}}} \cr & {\text{use }}\frac{1}{{{e^x}}} = {e^{ - x}} \cr & \int {\frac{{dx}}{{{e^x}}}} = \int {{e^{ - x}}dx} \cr & {\text{Setting }}u = - x,\,\,du = - dx,\,\,dx = - du \cr & \int {{e^{ - x}}dx} = \int {{e^u}\left( { - du} \right)} \cr & = - \int {{e^u}du} \cr & {\text{Integrating }} \cr & = - {e^u} + C \cr & {\text{substituting back }}u = - x \cr & = - {e^{ - x}} + C \cr} $$