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JH
Numerade Educator

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Problem 64 Medium Difficulty

Evaluate the integral.

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

Answer

$\frac{4}{3}(\sqrt{\sqrt{x}+1})^{3}-4 \sqrt{\sqrt{x}+1}+C$

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Video Transcript

Let's start off here by doing the use up. Let's just take you to be the entire denominator. Then here we will have to use the general from calculus. Teo, do the derivative, so I do the outer one first. So I'm using the power ruler. But that I have to go and also take the derivative of the inside. So here, let me I guess Let me write it as one over to radical X. That's the derivative but the inside. So we're all here. We're using the Kane rule. And to make this problem a little easier I'd like to re write all this in terms of you. So this is just one over to you Times one over to and I'll have to fill in this blank here and then the x So I need two for just rewrite radical exterior. So I'll just get that from this equation to square both sides. It's a practice one. So you just get you squared minus one instead of radical X and was good. And take this equation here and solvent for DX. So you have d X equals two times two is four we have Are you here and then you squared minus one. Do you? So now, after using our use up, we have one over you. And then we have R D X, which was for you. You squared minus one. Do you? Let's pull off the four. Cancel those use. We have you squared minus one, So just generate this. Using the power rule. Don't forget your constant of integration. See? And then the last appears to just replace you back in with a radical. So we have here you to the third power and then minus four times you and that's your final answer.