Problem 21 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int \arctan \sqrt{x}\ dx $

Answer

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Video Transcript

Let's try a U sub for this integral. Let's take you to be square root of X so that do you one over to X so we can rewrite this as t U equals one over to you dx and then just saw that for DX. So this is what will replace this term over here dx within the next in the raw. So this integral equals ark tan of you and then here for D s. We have to you, do you from this equation over here. So now is pull into two war and then let's use integration by parts. So here, let's use a different letter than normal. Let's use w let that be ten inverse of you then DW one over one plus you square, do you? Then we have a TV must be was left over, which is you. Do you then integrate that to get use were over too. So here we're using the formula. So we have to and then recall the buy parts formula in this case because I'm using the letter w So here, let's go ahead and multiply out this to to both herbs. So we'LL have you swear ten in verse. You, after cancelling out two twos minus integral use, were over one plus you squared and then do you? So now we have another integral to deal with here. So perhaps the easiest way to do this one is to do polynomial division. And then you can rewrite this. After doing the long polynomial division, you can go ahead and write. This of the quotient is one and the remainder is minus one, so we can rewrite it. Is this so let's go to the next page. And then we have one minus one over one. Plus he square and this is a much easy, easier in a role to do. There's no need to do partial fractions here. This already is a partial fraction. So for this integral, you may remember this one. Well, we actually use this fact already the derivative of ten in verses, one over one plus u squared or if you completely for about this fact, you can just do a trick sub. In either case, this hole is fresh in here and then we have minus you. And then, because of this double minus plus ten in verse, you and then go ahead and add that constancy and recall that we defined you to be scared of X. So let's go ahead and chlo that in here. So we have used square, which is X here. Technically, you should put absolute value. But in order for this to even make sense, this definition appear we need X to be bigger than or equal to zero. So we could just replace you square with X. So no absolute value necessary there. And then here we have square X, Linus, Rolex Plus ten in an ember through X and then plus e. And then here you could go ahead and factor out Ah, and first hand if you like, and that's a final answer.