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JH
Numerade Educator

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Problem 67 Medium Difficulty

Evaluate the integral.

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

Answer

$\frac{2}{3}\left[(x+1)^{3 / 2}-x^{3 / 2}\right]+C$

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Video Transcript

Let's go ahead and multiply. It's happened bottom by the conjugated. So let's write this So this first part will stay in the same and then I'LL multiply its happened bottom by X plus one minus radical X. This is a common trick. I'm often simplifies things. So after multiplying viso in the numerator, you just multiplied by one. And then in the denominator, we have a difference of squares that something of this form. And when you multiply that out, you always get a square minus B square. So on our problem that she's going to be X plus one that's a swear and then minus piece where? Which is just X. So here, of course, I'm taking eh to be radical X plus one B equals radical X. Now let's just simplified this denominator here. That's just simplifies toe one. And then I could just bright this as X plus one radical X t X. So if this plus one is bothering you, go ahead and do a U substitution before he do the power rule. But in any case, after integrating this, we use the power rule. So that becomes X plus one to the three halfs power, and then you multiplied by two over three and then my tiss and then hear this becomes X to the three half power to over three plus e, and that's your final answer.