Problem 17 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{x}{\sqrt{x^2 - 7}}\ dx $

Answer

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Video Transcript

let's evaluate the integral of X over the square root of X squared minus seven. So, looking in the denominator, we see an X squared minus seven, which is an expression of the form X squared, minus a square. And we know that in this situation the substitution to use is X equals a seek and data. Now, since a square to seven that tells us that we could take a to be the square root of seven. So our trick some should be X equals square root of seven c can data and taking the differential on each side. This gives us the X equals Route seven c can't time standard. So let's go ahead and plug in the stuff into the new integral. Also, maybe we should, before we do that, looking at the denominator, the square root. Let's go ahead and simplify that on the side. X squared minus seven. So X squared becomes seven. C can squared, and then we could pull out of seven. Factor it out. Then I could pull out the radical seven, and I can write C can't squared minus one as tangent squared, from which we have route seven times. Tan data. So this is our denominator. So coming back to the integral makes more room here. Okay, we have X and the numerator. So X's Route seven. So you can and then we have DX, which is route seven. Time's running out a room here. Let's come down here. C can't data tanta d data. So this is my numerator and then the denominator. We already simplify this. Yeah, Route 17 data. And then here we should cancel as much as we can. We see that we could cancel one of those roots, Evans. Also, the tension will go away and have this route seven up here. Let me go ahead and pull this out, okay? In front of the integral. And then we're left over with second time Second. So we have Seacon Square data detailer. Mhm. And this is one of the common in the rules that we've seen. We know the integral of this is just a tangent and at our constancy of immigration. So now at this point, we'll need to use a triangle because we want to express tanned data in terms of the original variable X. So we'll need a triangle here, So let's go to the next page. So we call our tricks up. X equals Route seven. Seek and data. Now, let's draw any right triangle that satisfies this property sloppy here. Okay, so let's go ahead and call this data over here. Now, If seeking of data is equal to X over route seven, let's go ahead and take the hype on Needs to be X and the adjacent side to be Route seven. Then if we choose these as our sides, then this equation appear is true. And then we could use Pythagorean theorem to find the remaining side each by Pythagorean theorem, we have this equation and then solving for age, we have radical X squared minus seven. Now going back to our original answer It was Route seven Tan Data plus E. Now we can go ahead and evaluate tangent potato. So we know tangent is opposite over adjacent. So we have route seven times H, Which is this over the adjacent room seven. Here. We see that we could cancel the Route seven's and we got our final answer Radical X squared minus seven plus c. And there's our answer