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Watch the following video to see the worked solution to this example.
For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.
Each inverse hyperbolic function has specific domain and range restrictions
Derivatives of inverse hyperbolic functions are found using implicit differentiation
Evaluate the following derivatives:
ddx(cosh−1(3x))ddx(cosh−1(3x))
ddx(coth−1x)3
ddx(cosh−1(3x))=3√9x2−1
ddx(coth−1x)3=3(coth−1x)21−x2
Watch the following video to see the worked solution to this example.
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This integral suggests a u-substitution. Let u=x2+9.
Then du=2xdx, or xdx=12du.
Rewrite the integral in terms of u:
∫x√x2+9dx=12∫1√udu
This is a standard integral. We get:
12∫1√udu=12⋅2√u+C=√u+C
Substitute back x2+9 for u:
√x2+9+C
However, we can express this result using an inverse hyperbolic function:
√x2+9=3√1+(x3)2=3sinh(sinh−1(x3))
Therefore, ∫x√x2+9dx=3sinh(sinh−1(x3))+C.
Applications of Hyperbolic Functions
The Main Idea
Catenary Curves:
A catenary is the curve formed by a hanging cable or chain under uniform gravity
Mathematically modeled by: y=acosh(xa)
Arc Length Formula:
Used to calculate the length of a catenary
Arc Length=∫ba√1+[f′(x)]2dx
Hyperbolic Function Properties:
1+sinh2x=cosh2x
This identity is useful in simplifying arc length calculations
Assume a hanging cable has the shape 15cosh(x15) for −20≤x≤20. Determine the length of the cable (in feet).
52.95ft
A suspension bridge cable forms a catenary with the equation y=50cosh(x50), where x and y are measured in meters. The cable is anchored at x=−100 and x=100. Calculate the length of the cable.
We use the arc length formula: Arc Length=∫ba√1+[f′(x)]2dx