Problem 51 Medium Difficulty

Make a substitution to express the integrand as a rational function and then evaluate the integral.

$ \displaystyle \int \frac{dx}{1 + e^x} $

Answer

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Video Transcript

Let's try the substitution you equals Eat of X. The goal here is to use a substitution to rewrite the inner grand one over one plus needed the X as a rational function. So taking this to the U Do You is here The X T X, which I could write, is you DX and then divide by you. Take it, do you over you equals DX. So in the original integral, I can replace DX with do you over you and then in the denominator, we just have one plus you. So let's clean this up a little bit. This is just one over and then we have you one plus you to you. And that's a rational function that they asked for. So taking this rational function, let's go ahead and do partial fraction to composition, using what the author calls case one you distinctly of actors. Let's go and multiply both sides by that denominator on the left, and then let's go ahead and simplify by pulling out of you from the left hand side. The constant term is one on the right hand side. It's a so those are equal and because on the right we have a plus B in front of the U next to the U. But on the left, there is no you. So the coefficient in front of you, you must be zero. So solving this for me, we get B equals negative one and then we could plug These values in for Ambien are partial fraction and then we take this and this is the term that we're replacing the fraction with. And then we integrate. Let's go to the next place to do that. They was one so integral over you, minus one for B. So we pull out the minus one plus you, do you the first general natural log? Absolute value. Second rule, absolute value. One plus you. Now that one Plus he was bothering. You feel free here to do a use up. Let's say w equals you plus one, and that should help you evaluate the integral with that said the last step here. Replace you with how it's to find needed X. You could drop the absolute value here since you two, the exes always positive. And you could also drop it here because either the X plus one is positive, the last possible thing we can do here is to use the fact that Ellen X in need of X r inverse functions. So remember, if you have in Versace and you composed them together, you always get X. And that's exactly what we have here. Natural Log is being composed with either the ex, so that is just X. And then everything else is the same, and that's your final answer.