750
Chapter 6 | Applications of Integration
d dx
f ( x )
f ( x )
1 1+ x 2
sinh −1 x
1 x 2 −1
cosh −1 x
1 1− x 2
tanh −1 x
1 1− x 2
coth −1 x
−1 x 1− x 2
sech −1 x
−1 | x | 1+ x 2
csch −1 x
Table 6.4 Derivatives of the Inverse Hyperbolic Functions Note that the derivatives of tanh −1 x and coth −1 x are the same. Thus, when we integrate 1/ ⎛ ⎝ 1− x 2 ⎞ ⎠ , we need to select the proper antiderivative based on the domain of the functions and the values of x . Integration formulas involving the inverse hyperbolic functions are summarized as follows. ∫ 1 1+ u 2 du = sinh −1 u + C ∫ 1 u 1− u 2 du = −sech −1 | u | + C ∫ 1 u 2 −1 du = cosh −1 u + C ∫ 1 u 1+ u 2 du = −csch −1 | u | + C ∫ 1 1− u 2 du = ⎧ ⎩ ⎨ tanh −1 u + C if | u | <1 coth −1 u + C if | u | >1
Example 6.49 Differentiating Inverse Hyperbolic Functions
Evaluate the following derivatives: a. d dx ⎛ ⎝ sinh −1 ⎛ ⎝ x 3 ⎞ ⎠ ⎞ ⎠
2
⎛ ⎝ tanh −1 x ⎞ ⎠
d dx
b.
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