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JH
Numerade Educator

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Problem 55 Medium Difficulty

Evaluate the integral.

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

Answer

$$2 \ln \sqrt{x}-2 \ln (1+\sqrt{x})+C$$

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Video Transcript

Let's try to take you to be square X. That's our use of motivated by this determined the denominator two to you. And we could write this as one over to you DX. So go ahead and solve this equation here for DX. So multiply that to you to the left. So also we have you squared equals x So this integral We have the X up here, so use this and then here X thought use where? Plus, he's where times you. So go ahead and cancel out. Ah, you And then what should we do here? See if that denominator factors If you'd like to use parcel fractions so we do And some recall you can rewrite this into grant in the form a over you be over you plus one. But then we have to go ahead and find a and e so here for and be we'LL have one for a minus one for Bea Evaluate those and then go back to the definition of you or, if you like, doesn't matter which however you like to write it, let's actually right in this form one half. So we have to Ellen exit the one half you could drop out flu values here because you was data that zero that here we have minus two natural log. Except the one half plus one. You could drop the bars here, too. And then here, we could simplify this. We're here. I'm using the property. That and therefore that's a final answer.