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}.
Answer and Explanation:
Become a Study.com member to unlock this answer! Create your account
View this answerLet'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...
See full answer below.
Learn more about this topic:
from
Chapter 13 / Lesson 5Explore the steps in integration by substitution. Learn the importance of integration with the chain rule and see the u-substitution formula with various examples.