Example 7.2.4(e): Applying the Substitution Rule

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Now it gets interesting again. In a previous example we had the square root of r² - x², and we used the substitution x = sin(t) because of the equality sin²(t) + cos²(t) = 1.

For the first integral we use the substitution

x = sinh(t) [ = 1/2 (e^t - e^-t) ] so that
dx/dt = 1/2 (e^t + e^-t) = cosh(t) or dx = cosh(t) dt

because of the hyperbolic trig equation

cosh²(t) - sinh²(t) = 1

(which you should verify). With that substitution we have that:

where arc sinh is the inverse function of sinh. Note that it is not hard to show that

Evaluating the second integral is left as an exercise (try the substitution x = cosh(t)).

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007