Find the integral {eq}\int \frac{1}{1 + e^x} \, \mathrm{d}x {/eq}.

Question:

Find the integral {eq}\int \frac{1}{1 + e^x} \, \mathrm{d}x {/eq}.

Integrals:

By now we have a number of methods and rules that we can take advantage of whenever we need to evaluate an integral. Evaluating the one above will require us to use a couple tricks. We will first use a substitution to simplify the integrand. Then we will use partial fractions (don't worry, it's pretty obvious), and lastly, convert the result back to being inb terms of {eq}x {/eq}.

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Let's write

{eq}\begin{align*} u &= 1+e^x \\ e^x &= u-1 \\ e^{-x} &= \frac1{u-1} \end{align*} {/eq}

Then

{eq}\begin{align*} du &= e^x\ dx...