Lecture 13.pdf - LECTURE 13 HYPERBOLIC FUNCTIONS II PO-NING...

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LECTURE 13: HYPERBOLIC FUNCTIONS II PO-NING CHEN In this lecture, we will look at more integrals involving x 2 - 1 and x 2 + 1. Then we will look at partial fraction decomposition which is used to integrate rational functions. 1. More examples of Hyperbolic Functions and their properties The hyperbolic cosine cosh x and the hyperbolic sine sinh x as follows: cosh x = e x + e - x 2 sinh x = e x - e - x 2 In the following, we will verify that they satisfy the following properties d dx cosh x = sinh x d dx sinh x = cosh x cosh 2 x - sinh 2 x =1 cosh 2 x = cosh 2 x + sin 2 x sinh 2 x =2 sinh x cosh x For integrals involving x 2 + 1, we will use x = sinh u and for integrals involving x 2 - 1, we use cosh u . Exercise 1 Compute Z p x 2 - 1 dx using x = cosh u . Answer to Exercise 1 Let x = cosh u then dx = sinh udu and 1
2 PO-NING CHEN Z p x 2 - 1 dx = Z sinh 2 udu = 1 2 Z cosh(2 u ) - 1 du = 1 4 sinh(2 u ) - 1 2 u + c. We write it back to x using the double angle formula: 1 4 sinh(2 u )+ u + c = cosh u sinh u 2 - u 2 + c = 1 2 [cosh u sinh u - u ]+ c = 1 2 [ x p 1 + x 2 - cosh - 1 x ]+ c Exercise 2 Compute Z 1 1 + x

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