Chapter 6 | Applications of Integration
747
d dx (cosh
x ) 2
b.
Solution Using the formulas in Table 6.2 and the chain rule, we get a. d dx ⎛ ⎝ sinh ⎛ ⎝ x 2 ⎞ ⎠ ⎞ ⎠ =cosh ⎛ ⎝ x 2 ⎞ ⎠ ·2 x
d dx (cosh
x ) 2 =2cosh x sinh x
b.
6.47
Evaluate the following derivatives:
⎛ ⎝ tanh ⎛ ⎝ ⎜ 1
⎛ ⎝ x 2 +3 x ⎞ ⎠ ⎞ ⎠
d dx d dx
a.
⎞ ⎠ ⎟
b.
(sinh x ) 2
Example 6.48 Integrals Involving Hyperbolic Functions
Evaluate the following integrals: a. ∫ x cosh ⎛ ⎝ x 2 ⎞ ⎠ dx b. ∫ tanh xdx
Solution We can use u -substitution in both cases.
a. Let u = x 2 . Then, du =2 xdx and ∫ x cosh ⎛ ⎝ x 2 ⎞ b. Let u =cosh x . Then, du = sinh xdx and ∫ tanh xdx = ∫ sinh x cosh x ⎠ dx = ∫ 1
2 sinh ⎛
⎞ ⎠ + C .
⎝ x 2
udu = 1
u + C = 1
2 cosh
2 sinh
dx = ∫ 1
u du = ln| u | + C = ln | cosh x | + C .
Note that cosh x >0 for all x , so we can eliminate the absolute value signs and obtain ∫ tanh xdx = ln(cosh x )+ C .
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