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Use the hyperbolic substitutionn $x=\cosh u,$ the identity$\sinh ^{2} u=\frac{1}{2}(\cosh 2 u-1),$ and the results referenced in Exercise 51 to evaluate$$\int \sqrt{x^{2}-1} d x, \quad x \geq 1$$

$\frac{1}{2} x \sqrt{x^{2}-1}-\frac{1}{2} \cosh ^{1} x+c$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

PRINCIPLES OF INTEGRAL EVALUATION

Section 4

Trigonometric Substitutions

Integrals

Integration

Integration Techniques

Trig Integrals

Trig Substitution

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University of Nottingham

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we're going to evaluate the hyperbolic co sign of X squared minus the hyperbolic sign of X squared. I bring the definitions of these two above. So what? Substitute those definitions in. So we get the hyperbolic co sign of X squared minus and they provide a sign of X where is equal to need to the X plus e to the negative X over two squared minus. He'd to the X minus e to the negative X over two square. So this is equal to well, for each term we have a 1/2 squared, so we can just factor that out so 1/4 times. Okay, so we have e to the X plus e to the negative X squared, which is equal to e to the x times e to the x or d to the two x plus two times e to the x times e to the negative x e to the x times E to the negative. X is just one. So we have plus two plus e to the negative x times E to the negative X, which is equal to e to the negative two X then we have minus you to the X minus e to the negative X squared. So first we have you to the X Times need to decks, which is eat the two x minus two times e to the x times each of the negative X which again is just going to be too. So we have minus two plus negative eat a negative x times negative e to the negative X which gives us plus need to the negative two x So in these parentheses we have eat to the two X minus e to the two X So these cancel. We have two minus negative too. So we're going to have a four in there and then we have eaten the negative two x minus e to the negative two x So those canceled as well. So this is equal to 1/4 times for which is equal to one. So the hyperbolic co sign of X square minus the hyperbolic sign of X squared is equal to one

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