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Evaluate the integral using integration by parts with the indicated choices of $ u $ and $ dv $.

$ \displaystyle \int \sqrt{x} \ln x dx $ ; $ u = \displaystyle \ln x $ , $ dv = \sqrt{x} dx $

$\int \sqrt{x} \ln x d x=\frac{2}{3} x^{3 / 2} \cdot \ln x-\frac{4}{9} x^{3 / 2}+C$

Integration Techniques

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Nhon M.

October 5, 2017

Hi there

Okay, So in this problem we're asked to evaluate the integral using integration My parts with indicated choices of you and Devi. So normally when you're given a problem like this, you're not given the you and Devi to use for your integration by parts You have to figure it out Experiments so on so forth In this case, they were making your life easier By showing you the way for say so to do intubation parts You gotta recognize what you you and what your DV parts are. So you have a general you know you're giving your problem squared of axe National of X d x You want to like he compose it into its U N D V parts here underlined the you part which is natural backs and the DV part escort of X t X So I reorganize it as natural of x times at skirt of X t x which is the unique Devi. The general formula for education by parts is integration of you Devi musical to you Be honest in the grill of the EU All right, so we know what you and DVR because we're given them what we have left to figure out are the V and D you So I'm gonna lips, It's a bee And then there's you lips. I cannot miss that. The color coding there. But yet the point, hopefully so to figure out what b and d ur we set up this table yuk was not. Charlotte backs Devi equal skirt of back to the X or extra the one half of the ex like I prefer that form because it's easier to integrate. DuSable toe one one over x d x. This is the derivative of natural log of X, and the is two thirds after three halves, because when you integrate actually one half you have to add one to the one half say of three halves and you multiply. Marry circle reciprocal. So it's two thirds times except we have. So our general formula becomes into of national of X times squared of actually X is equal to natural. Kovacs was just the you time's two thirds except the rehabs, which is your V minus the integral of two thirds three halves, which is your V times one over X t X, which is your due you? No, sir, I'm not adding see yet to the V because it's ah, it's the middle of an interrogation by parts. It's a middle step of the new Asian apart. So we're only adding, See at the end, basically. So here we're promised, almost done we have all we have left to do is clean the notations up. So, for example, the two natural log of X times tooth There is excellent rehabs. It becomes two thirds extra three house naturally mix. I write thie left hand side to what it was originally which is controlling squared of axe Natural log of X t x, The right part I'm here at drill of two thirds extra three halves. What? I was one of the acts of the acts. If you multiply it out X Earth rehab stamps one of her exes as three halves times. Excellent native one which you get extra one half and I pull out the two thirds outside of the integral. So you got two thirds times in a row of X is one half dx in the and then you further clean it up. You confusion solve this and you go now, eh? So you got two thirds times actually three have natural good max plus two thirds times another two thirds times X with rehab because that's the integral of X one half plus e. So you have equals to two thirds, actually. Three halves natural log of X plus four nights as two three halves plus C.