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Make a substitution to express the integrand as a rational function and then evaluate the integral.

$ \displaystyle \int \frac{1}{\sqrt{x} - \sqrt[3]{x}}\ dx $ [Hint: Substitute $ u = \sqrt[6]{x} $.]

$$2 \sqrt{x}+3 \sqrt[3]{x}+6 \sqrt[6]{x}+6 \ln |\sqrt[6]{x}-1|+C$$

Integration Techniques

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but he is the given substitution to rewrite the Inter Grant as a rational function. So if this is our substitution, let's go ahead and raise both sides to the sixth power. Then take a differential on the right, which is get DX. So this tells us that we can replace DX and the inside rule with this expression here. And then we should just go ahead and rewrite the's terms in the denominator in terms of you. So school relax except the one half, and that's equal to, let's say, X three over six and this is X to the one over six to the third power so that you two the other power fire substitution. Similarly, Cuba Rex Okay, except the one third except the two over six. So this X to the one over six and then square. So that's you use weird. So, using these faxed, we have on top just the X that sits you to the five to you and on the bottom Rue X. We found that to be you Cube. So there's are you Cuban in the bottom and then cube root of X. We found that to be you square. We see in the denominator, we can factor out he's square. And then we could go ahead and cancel that you squared with two of these up top and we're just left over with three. So we go in and pull out that six you cubed and then you minus one on bottom. Now, this is a rational functions that they wanted. Here we see that the numerator has larger degree. Let's go ahead and do long division polynomial division. Somebody write that over here on the right. You minus one going into you to the third power and you goes into you, Cube. So we have you squared up here and then multiply and then subtract. We have you squared. So we have a you up here and then use were mine Issue subtract running out of room here that's going to leave us with a U And then you goes into you one time, and then we have you might have one in that equals one. So that's our major. That means that we could come over here. We have our integral and then the quotient wass you swear. Plus you plus one. Then we have our remainder over the original denominator. Now, if you got an injury, use the power rule for the first three terms. Sixty square over too. Plus six. You. And then for the last term, I still should have a six here. Six natural log. Absolute value. You minus one. Okay. Plus E. And if it helps you, it might help you to do use up here. You could let w bu minus one if that minus one is bothering you. So the last step here is to just simplify the fractions and then replace you with X, using the original substitution that was suggested in the head. So let's go to the next page to write this. We just have to and then you cubed. That's just squared of X. And then we have a six over to his three and then use weird. We saw that to just be the cubano vexed. And then we have six u survivors are substitution. That's just the six. Relax and then plus six natural log. And then we have absolute value. You minus one. So you want you could I'm being inconsistent here. I should maybe be consistent. Let me go back and write. This is six foot of lex minus one and then plus or constancy. And that's your final answer.