What is the integral of inverse e^x?

integral of inverse e^x

The integral of

e^{-x}

-e^{-x} + C

.

Solution. We want to determine the integral of

e^{-x}

, i.e.:

\begin{align*} \int \frac{1}{e^x} dx = \int e^{-x}dx. \end{align*}

We will use the substitution method: let

u = -x

, then

du = -dx

. Then we get the following integral:

\begin{align*} \int e^{-x} dx &= - \int e^u du \\ &= -e^u + C \\ &= -e^{-x} + C. \end{align*}

Therefore, the integral of

e^{-x}

-e^{-x} + C