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JH
Numerade Educator

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Problem 44 Medium Difficulty

Evaluate the integral.

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

Answer

$2 \sqrt{1+e^{x}}+\ln \left(\sqrt{1+e^{x}}-1\right)-\ln \left(\sqrt{1+e^{x}}+1\right)+C$

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Video Transcript

Let's start this one off with the use of Yeah. Then from here we have Do you? So this is using the chain rule from calculus. Mhm. And then we can go ahead and rewrite this. So how do we rewrite E in terms of you? Square both sides and subtract one. And so that's not a negative. Let's erase that. There we go. So either the X wicked right is you squared minus one. And then here we have two times you. So looking at this equation here, it's got in. Get the X by itself. Okay, so now rewrite this in overall. Yeah. This first term right here. The square root, That's just you. And then DX. We use this over here, and there we go. So let's pull out that too. You square, you squared minus one. Because both of these polynomial is have the same degree. We would need to do polynomial division here. So let's do that division, and then we end up with one, and then a remainder of one. So then looking at this fraction here, you may have memorized this already. If not just right. That is one over. You plus one u minus one partial fraction. Yeah, So go ahead and solve that. For A and B, we end up with a equals negative. One half B equals one half. So let's go to the next page to plug that in. And then we have Yeah. Oh, so let's integrate this to you. There's our natural log and then another natural log. Cancel out the one half into two and your c and then here recall the definition of you. Oh, so here we have two times the radical minus natural log. And then here you plus one here you could drop the absolute values because the square root plus one is always positive. And here we can also drop the absolute value if we want to write this as one plus e to the X minus one because one plus eat of the eggs is always bigger than one. So this implies the square root of one Placido X is also is bigger than the square root of one. Therefore, when we subtract, this is a positive number, so we can drop the absolute value If we write that adding your constancy. And that's the final answer