Problem 3 Medium Difficulty

Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle.

$ \displaystyle \int \frac{\sqrt{x^2 - 4}}{x}\ dx $ $ x = 2 \sec \theta $

Answer

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Video Transcript

here. We like to evaluate the integral using the indicated shrink. So here the tricks of his X equals to thicken, from which we conclude D X is to seek and data can dated data so we can rewrite this General. This becomes square root of X Claire so that X squared becomes foresee can square and then DX is to seek and data Santiago de Data. And in the Dominator we just have X which was to seek antenna. So already we see that we can cancel off these two thickens and then also before we go on, let's simplify this radical here so we could factor out a four there. So taking four out the square and becomes a too. And then we have sequence where they don't minus one, which is too radical chance for data, which is too thin data. So we have integral toothy and data time Stan Data and we can rewrite tangent squared, which is what we have here. Tan square Some first let's pull up the two and then we could write chance. Where is sequence where data minus one. And we know the inside derivative of seek and squared is tangent and then integral of one is just data. So we have a minus two data, plus he And now this is the point where we can use the right triangle to try to simplify further. So according to our original substitution, we have seek and data equals X over two. So our triangle may look something like this, right? Triangle. Here's data and since the sea can is except for two, we have two of the corresponding sides, and then we can use put agree and dirham to find the remaining side. Let's call it a JJ. We know that h flared plus two squared his X square, so h is equal to the square root of X squared minus four. So now we have all three sides of the triangle. So we have two times tangent of data. So attention of data. It's just opposite over adjacent, So H over too. So we have radical X squared minus four. That thing's divided by two on the outside of the radical minus two. And then they'd know we really can't get the value of data from the triangle or we can. But it's not necessary. You could just use this equation appear to find data. So take the university can on both sides and we get data is seek an inverse of X over too. So we're basically finished The last thing to do here, which I'll write up. It's just a cancel off these twos in the first expression. So I'll need to go to a new page for this. So we have square root, no explore minus four minus two. Seek and embers X over, too. Plus he and there's our answer.