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Numerade Educator

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Problem 32 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int^{4}_{1} \frac{\sqrt{y} - y}{y^2} \,dy $

Answer

$1-2 \ln 2$

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Video Transcript

Our task is to find the integral from 1 to 4 of route. Why minus Y. All over Y squared, do I? Uh so I would have my students rewrite this still the integral from 1-4. And still dy but a trick of doing this is to divide each piece by the monem you'll remember the square root is the same thing as to the one half power. So you can subtract those exponents. One half minus two Is -3/2. Um Still a minus, but then also divide the next piece. Uh Why did the first? Well, I would actually just leave that as one over. Why you could leave as Why did a negative first? Let's actually do that real quick. Because if you follow your rules for the anti derivative, you know where you add one to the exponents and you multiply by the reciprocal of your new exponent. Um If you were to try that with this one and you add 12 negative and you get zero but zero doesn't have a reciprocal. So it makes sense that there must be a different derivative and had different derivatives. Natural look And that's from 1 to 4. So at this point you can plug in your upper bounds and from both ford and remember the the negative power makes this a fraction. So we're looking at negative two over because of the negative exponents and then it's a square root of four. So it'll be too minus Natural log of four and then you have to plug in your lower bound Here and here. What's nice about the lower bound though is one to any power is just one because who cares? It's a in the denominator divided by one square to one. So it's just negative two minus natural log of one. Now, here's the thing is I expect my students know without a calculator, natural log of one is zero. So that kind of goes away. Um, so as far as anything else goes, we're looking at -1. This turns us into plus two. So we're looking at positive one, Looking at the answer kids one and you can leave it as natural law before. Not sure why, but the answer key is correct. That you could change forward to be two squared. I'm just explaining why the answer key is what it is and then the power rule says you can move this in front. Um, It's a law of logs. The family's curious. That's why both of these answers are correct. So I'm circling them both. I'm telling you where I would stop, but you could keep going if you want to.