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Evaluate the integral.

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

$\int \sqrt{x^{2}+2 x} d x=\frac{1}{2}(x+1) \sqrt{x^{2}+2 x}-\frac{1}{2} \ln \left|(x+1)+\sqrt{x^{2}+2 x}\right|+C$

Integration Techniques

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here we have the integral of the square root of X squared plus two x. Let's go ahead and take this quadratic and complete this where So taking half of two, we get one. And if we square that we also get one, let's go ahead and add a one to this quadratic and then we make up for it by subtracting one. And now we've completed the square. We can write the quadratic and parentheses, X plus one square, and then we have a minus one. So here this is very similar to an expression of the form X squared, minus a slur. Where is one? So you, khun, basically treat it like it's of this form in this case, X squared, minus a squared. He was a xB. A C can't daito. So in our case, let's just take X plus one. The thing that's being squared, the variable that's being swear that's your ex. In this case, it's X plus one, and we said that equal toe a C can't. Data is one. So we just have seeking data differentiating inside. We get the exes seek and data tan data data. So after the first step, we would have completed the square. So we had this expression in the radical. Then after the tricks of X plus one square becomes seek and square data and D X becomes sick and data Xanterra Dictator. Now we know that C can't square minus one is tangents. Where so the square root of tangents Where is just simply tension? So we have tension data time seeking time standard. So now we have a trillion rule. One way to proceed here is toe replace tangent. I was seeking squared minus one. Then we can distribute and let's go ahead. Explain this up into two intervals. Now I'm running out of room here, So let me go to the next page Something rewrite what we had now these were both Trigon novels that were exercises in seven point three. So let me comment on the first Integral for this one, the way to proceed is integration. My parts take you to be seeking so that do is seek and data tan data and then take Devi to be sequence where? Data So that V equals Siegen data. So you'd use integration by parts for this integral And after doing so, you should obtain one half seek in time, Sam plus one half natural log, Absolute value seeking plus tan. So this sum is just for the first integral and the second, integral, it's in the table. It's one that you've come across many times at this point. So this one you may have just memorized and we have minus Ellen absolute value seeking data, Tristan Data. Plus, they are constant of integration. See, here we can combine this natural log with the other. So doing so. We have one half seek in time. Stan Gin minus one half natural log C Camp Liston Now at this point will rely on the triangle too simple by the rest. So let's go to the next page. So me rewrite what we had, okay. And our recall that our tricks of was X plus one equals he can. So if we want, we could think of this a Sikh and equals x plus one over one from their weekend draw a triangle. So C can is hi, Patton, who's over adjacent. So let's take this to be X plus one and the adjacent to be one. And then we could find this missing side each by pathetic grain. Sterile, a squared plus once where equals X plus one square. So subtract one and take the square root and you have each. If you want, you may write it this way. You might got into simplify that if you like, This becomes X squared. Plus two experts are equal to each other, so it really doesn't matter there. So now, coming back and evaluating these trade functions, we have one half seeking. We know seeking is just X plus one time's tangent. So tangent is H over one. So it's just a JJ. So this is radical X squared, plus two eggs. And then we have minus one half natural line, absolute values. He can't, which is X Plus one and then plus ten. What is radical X squared Plus two X. And then we put her constancy after this, and there's a final answer