748
Chapter 6 | Applications of Integration
6.48 Evaluate the following integrals: a. ∫ sinh 3 x cosh xdx b. ∫ sech 2 (3 x ) dx
Calculus of Inverse Hyperbolic Functions Looking at the graphs of the hyperbolic functions, we see that with appropriate range restrictions, they all have inverses. Most of the necessary range restrictions can be discerned by close examination of the graphs. The domains and ranges of the inverse hyperbolic functions are summarized in the following table. Function Domain Range sinh −1 x (−∞, ∞) (−∞, ∞)
⎡ ⎣ 0, ∞)
[1, ∞)
cosh −1 x
(−1, 1)
(−∞, ∞)
tanh −1 x
(−∞, −1) ∪ (1, ∞)
(−∞, 0) ∪ (0, ∞)
coth −1 x
⎡ ⎣ 0, ∞)
(0, 1]
sech −1 x
(−∞, 0) ∪ (0, ∞)
(−∞, 0) ∪ (0, ∞)
csch −1 x
Table 6.3 Domains and Ranges of the Inverse Hyperbolic Functions
The graphs of the inverse hyperbolic functions are shown in the following figure.
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