# Lecture 12 .pdf - LECTURE 12 HYPERBOLIC FUNCTIONS PO-NING...

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LECTURE 12: HYPERBOLIC FUNCTIONSPO-NING CHENIn this lecture, we will introduce the hyperbolic function and apply it to compute theintegral involvingx2-1 andx2+ 1. Here are examples of the integrals we will learn:Zpx2-1dxZpx2+ 1dxZ1x2-1dxZ1x2+ 1dxThe hyperbolic cosine and the hyperbolic sine are a pair of functions which share manyproperties of the usual cosine and sine functions that are useful in trigonometric integralsand trigonometric substitution. Here are the list of properties.ddxcosx=-sinxddxsinx= cosxcos2x+ sin2x=1cos 2x= cos2x-sin2xsin 2x=2 sinxcosx1.Hyperbolic Functions and their propertiesWe will define the hyperbolic cosine coshxand the hyperbolic sine sinhxas follows:coshx=ex+e-x2sinhx=ex-e-x2In the following, we will verify that they satisfy the following properties1
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