[tex]\displaystyle\int e^{-x}\,\mathrm dx[/tex]
You probably know that the antiderivative of [tex]e^x[/tex] is simply [tex]e^x[/tex]. So to get something that resembles this form, let [tex]u=-x[/tex]. Then [tex]\mathrm du=-\mathrm dx[/tex], or [tex]-\mathrm du=\mathrm dx[/tex].
Now the integral is
[tex]\displaystyle\int e^{-x}\,\mathrm dx=\int e^u(-\mathrm du)=-\int e^u\,\mathrm du=-e^u+C=-e^{-x}+C[/tex]
