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

$ \displaystyle \int \frac{dx}{x^2 + x \sqrt{x}} $

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

Integration Techniques

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let's evaluate the integral by first using a substitution and then possibly using partial fraction the composition for the rational function. So here, let's take you to be squared of X, which we can go ahead and write as you squared equals X and then taking a differential on the side to you know, do you equals the X. So could we can replace DX and the answer girl with to you do you So this inner world becomes to you, do you up top on the bottom. We see it X square there. So let's take this equation here and square both sides So that means X squared. I could replace with you to the fourth power and then we see an X that's just use where and then we see square affects that touches you scored and simplify this our new integral, which is not a rational function. We can take a UFO. We can factor out to you in the denominator and we could even cancel you hear. So now we have to one over you square you plus one to you. Now let's go ahead into a partial fraction to composition on our rational function. So we have one over you squared you plus one that should be of the form over you be over you square. So this is what the author calls case too. For the first time that say you square, that's a repeated linear factor you and then we have a case One. This is a non repeated linear factor. So we just get see you plus one taking our new equation Let's go to multiply both sides by this denominator and the left So we have one equals Hey you. Then you plus one be your plus one and then we have See you, slur It's good. Maybe simplify this. Expand the right hand side and then combine like terms Here let's pull out of you square a policy It's part of you a plus b and then we're left over with B And this is where we'LL get our system of equations on the right. We have a policy in front of the U Square but there is no use square on the left. That means a policy of zero similarly a plus B a zero and then we must have be peoples want the constant terms. So that's a three by three system Let's call in the next patients on this. So we had b equals one A plus e equal zero a plus B equals zero. So go ahead and solve this for a you have a equals negative. B b was one. So we get a negative one here and then C equals negative, eh? But is negative one. So she is one. So now we have a B and C Let's go ahead and plug in those values a, B and C into our partial fraction to composition. So we had previously pulled a tuo in front of the integral. Now we have a over you b which was won over you square and see which was also won over you plus one. She had one of their army go back. I'm starting to like it too. One over you plus one, Do you Now for the second interval, we could use the power rule And if this he plus one is bothering, you could do a use up here. So go ahead and evaluate those three for the first inter girl. Negative two natural log absolute value you And then for the next one, we get negative to over you using the power rule and then for the last one to Ellen you plus one. Plus he now we could go ahead and replace you with how it was defined. Recall we defined in the original in the beginning of this problem. So it's going to replace you. And instead of writing squared of X, let me right next to the one half year and we'LL see why in a second and then we have to Ellen and then squared of X plus one. And here I'll drop the absolute value because one plus square root of X is always bigger than zero and then plus e. So the last thing you could possibly do here is rear a Ellen X one half good. We could even take the one half out of the absolute value and then right, this is one half Ellen necks. So when we do that and our problem, the one half will cancel with the two. So we just have negative natural log absolute value minus two over radical X plus two natural log radical eggs plus one plus e. And that's a final answer