How to evaluate the following definite integral? $$\int_0^{\ln3} e^{x-e^x}\,\mathrm dx.$$
Should I use some sort of U Substitution?
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7$\begingroup$ Write the integrand as $e^x/e^{\color{maroon}{e^x}}$ and let $\color{maroon}{u=e^x}$. $\endgroup$– David MitraApr 26, 2014 at 22:32
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1 Answer
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Use the substitution $u=\mathrm{e}^{x}$, and $\mathrm{d}u=\mathrm{e}^x$, then we get \begin{align*} \int_0^{\ln3}\mathrm{e}^{x-\mathrm{e}^x}\,\mathrm{d}x &= \int_1^3\mathrm{e}^{-u}\,\mathrm{d}u \\ &=\left[-\mathrm{e}^{-u}\right]_1^3 \\ &=\frac{\mathrm{e}^2-1}{\mathrm{e}^3} \end{align*}
