Problem 5 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{\sqrt{x^2 - 1}}{x^4}\ dx $

Answer

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Video Transcript

we'd like to evaluate the integral off the square root of X squared minus one, all divided by excellent fort. So since the expression and the radical X squared minus one is of the form X squared minus a square where is one we'LL use the trip substitution of this forum. So in our case, we have X equals one times he can such a Sikh and data So that's how it reads. Taking a derivative gives us the X So let's plug this in so the integral equals. Also, before we go on, let's evaluate X squared minus one inside the radical So this will be C can't squared minus one. So squared chance where data which is simply tan data. So we have a tan data in the numerator so and the ex becomes seeking Taito Pantera And in the denominator, we have X to the fourth power. So that's a sequence of the fourth power. I couldn't rewrite this so we see that we can cross off one of the sea cans So we have Ah, we take that off and then we're left with C can't to the third power on the bottom. Some top. We have chance for data and the bottom C can't keep Daito. So at this point, we could use the fact that C can't. Data is one of her co signed by definition. So this becomes tan squared Data times Cho san, Keep Daito. So tan squared is science, squid, overcoats and square. So, in the way of times Cho sang cubed. Here we do more cancellation and we have science. Where? Data cosign data data. So I'm running on room here. Let's go to the next page. At this point, we have science for data. Cho, Scientific data Solitude is air U substitution, then do you? It's Cosa data data data. So we have you square, Do you? Which is you cubed over three plus e and then we back Substitute to write this a sign. Cube data over three. Plus he So this is the anti derivative. But we want to get back into the variable x, not Ada. So at this point, we go back to our tricks up, which was X equals C can terra, you could also think of this is saying she can't Data equals X divided by one. So when we go to the triangle. So seeking is high pound news over adjacent and that's X over one. And then the remaining side age we could find Bi put a great hero A squared plus one equals x square. So h is square room X squared minus one. Now we could find signed Ada by looking at the triangle. So sign of data we'LL be except her age or excuse me h of Rex. So we plugged this in. So we have one third square root of X squared minus one over x cubed plus e And then we could simplify this to get our final answer Oliver X squared minus one to the three halfs and the numerator Whoever three execute in the bottom Plus he and there's our answer.