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Find the average value of $ f(x) = \frac{\sqrt{x^2 -1}}{x} $ , $ 1 \le x \le 7 $.

$\frac{\sqrt{48}}{6}-\frac{\sec ^{-1} 7}{6}$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 3

Trigonometric Substitution

Integration Techniques

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

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Let's find the average value of the function square root of X squared minus one on divided by X. A CZ X goes from one to seven. So recall the formula for the average value from A to B, it's one of her B minus a times the integral of the function f from A to B and our problem is one and be a seven. So we have average value is one over six and a girl from one to seven X squared minus one and the radical on the bidet backs. And now we should use the tricks up for this for this in a rural. So looking at the numerator, we see X squared minus one in the radical. So we should take extra B one time. See, can't Daito. So is one here. So by the I mean the and the formula for the truth substitution not ta from the average value, which also happens to be one. Okay, so here the eggs seek and data tan data. So for this inaugural, we can go ahead and try to find the fate of values. But sometimes it's best to not do that because it might be hard to find when the values of data that make seek and data equals two seven. And so because of that, let's go ahead and rewrite this Since we won't, we don't care to use the data values because we're eventually going to go back into X Delicious to know the date of values. I see Andy So these are the limits of integration in terms of data necks Well and again, we don't need CND because we'll eventually back substitute X, in which case we could use one and seven is our end points. Okay, so X squared minus one. That sequence where minus one and then dx Siggins have Stan and the denominator is just x which is he can't So a few things we can do here we could cross off the sea cans and then notice that the radical equals radical tan squared. What is just San Diego? So we have won over six integral CD of town square and we could use the Pythagorean identity here to rewrite This is seek and squared minus one. But she'll be easier to integrate, so I should have a debate in here somewhere. So this is one over six and girls he can square this tangent and a girl One is just data Now we'LL use the right Rangel toe back substitute in terms of X So our tricks up was X equals C can We could also read this as seek and data equals X over one So seeking his hypotheses over adjacent and that's X divided by one. So let's put the X here No one here and we could find this h by using the Pythagorean zero. So by Pythagorean a squared plus one is x square. So we have a cheap ALS radical X squared minus one. And now that we have all three sides, we can go ahead and find tangent and data. So this is one over six. So tangent is h over one. So just X squared minus one and data we could find data from the tricks up taking Seek an inverse on both sides. You have thinner equals. C can't inverse of X. And now, since we're back in X, we could use our original limits one in seven. So let's go to the next page and simplify this So plugging in seven First, we have seven squared minus one in the radical. So that's forty eight and then plugging in seven. We have seek an inverse. So this is from plugging in seven first. And then when we plug in zero, we get one minus one minus c can inverse of zero. Seems we seek an inverse of one and seeking an inverse of one equal zero because he can use there was one one. So here we could ignore the second, sir, and we're left with our final answer. Radical forty eight over six minus to seek an inverse of seven over six, and that's it.

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