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Numerade Educator

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Problem 11 Easy Difficulty

Find the general indefinite integral.

$ \displaystyle \int \frac{1 + \sqrt{x} + x}{x} \, dx $

Answer

$\ln x+2 \sqrt{x}+x+c$

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Video Transcript

So there's actually two rules, you're supposed to know this problem that the anti grav of. Um Yeah. X. Oops, I did that wrong next to the end is equal to X. To the M plus one over. Uh Multiply by the reciprocal and policy. And the other rule is if you had a problem, that was just the integral of excellent negative first power, D. X. So it's one over X. If you add 12 negative when you're at zero, which is why we need to look at natural log effects. So these are the two themes in this problem. Uh And that works as long as we just have a Selmer difference of pieces here. So if one plus route X plus X. All over X. Dx. And so what I would have my students do is rewrite this problem as one over X plus. When you subtract experiments here, that would be X to the negative one half power. Um And then I guess I should clarify, I'm dividing each piece by X. And a lot of my students are really good at going backwards, like taking three separate fractions and then putting them together to look like that and for whatever is and they struggle to split up a fraction, but it is exactly the same. So the whole reason I'm doing that is the anti derivative of one of the X. As I mentioned before is natural log effects. And then plus I think it has to have absolute value here because X could be a negative. Um Mhm. Secretary plus see there's uh but when you add one to that experiment, make it one half plus one is one half times the reciprocal And then the anti growed up of one is just X. And then don't forget about plus C. And you can rewrite this problem in different ways, but this should be a good enough answer. Uh We'll move on.