6. Applications of Integration

# 6.9 Calculus of the Hyperbolic Functions

### Learning Objectives

• Apply the formulas for derivatives and integrals of the hyperbolic functions.
• Apply the formulas for the derivatives of the inverse hyperbolic functions and their associated integrals.
• Describe the common applied conditions of a catenary curve.

We were introduced to hyperbolic functions in Introduction to Functions and Graphs, along with some of their basic properties. In this section, we look at differentiation and integration formulas for the hyperbolic functions and their inverses.

# Derivatives and Integrals of the Hyperbolic Functions

Recall that the hyperbolic sine and hyperbolic cosine are defined as

The other hyperbolic functions are then defined in terms of and The graphs of the hyperbolic functions are shown in the following figure.

It is easy to develop differentiation formulas for the hyperbolic functions. For example, looking at we have

Similarly, We summarize the differentiation formulas for the hyperbolic functions in the following table.

Derivatives of the Hyperbolic Functions

Let’s take a moment to compare the derivatives of the hyperbolic functions with the derivatives of the standard trigonometric functions. There are a lot of similarities, but differences as well. For example, the derivatives of the sine functions match: and The derivatives of the cosine functions, however, differ in sign: but As we continue our examination of the hyperbolic functions, we must be mindful of their similarities and differences to the standard trigonometric functions.

These differentiation formulas for the hyperbolic functions lead directly to the following integral formulas.

### Differentiating Hyperbolic Functions

Evaluate the following derivatives:

#### Solution

Using the formulas in (Figure) and the chain rule, we get

Evaluate the following derivatives:

#### Hint

Use the formulas in (Figure) and apply the chain rule as necessary.

### Integrals Involving Hyperbolic Functions

Evaluate the following integrals:

#### Solution

We can use -substitution in both cases.

1. Let Then, and
2. Let Then, and

Note that for all so we can eliminate the absolute value signs and obtain

Evaluate the following integrals:

#### Hint

Use the formulas above and apply -substitution as necessary.

# 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.

Domains and Ranges of the Inverse Hyperbolic Functions
Function Domain Range

The graphs of the inverse hyperbolic functions are shown in the following figure.

To find the derivatives of the inverse functions, we use implicit differentiation. We have

Recall that so Then,

We can derive differentiation formulas for the other inverse hyperbolic functions in a similar fashion. These differentiation formulas are summarized in the following table.

Derivatives of the Inverse Hyperbolic Functions

Note that the derivatives of and are the same. Thus, when we integrate we need to select the proper antiderivative based on the domain of the functions and the values of Integration formulas involving the inverse hyperbolic functions are summarized as follows.

### Differentiating Inverse Hyperbolic Functions

Evaluate the following derivatives:

#### Solution

Using the formulas in (Figure) and the chain rule, we obtain the following results:

Evaluate the following derivatives:

#### Hint

Use the formulas in (Figure) and apply the chain rule as necessary.

### Integrals Involving Inverse Hyperbolic Functions

Evaluate the following integrals:

#### Solution

We can use in both cases.

1. Let Then, and we have
2. Let Then, and we obtain

Evaluate the following integrals:

#### Hint

Use the formulas above and apply as necessary.

# Applications

One physical application of hyperbolic functions involves hanging cables. If a cable of uniform density is suspended between two supports without any load other than its own weight, the cable forms a curve called a catenary. High-voltage power lines, chains hanging between two posts, and strands of a spider’s web all form catenaries. The following figure shows chains hanging from a row of posts.

Hyperbolic functions can be used to model catenaries. Specifically, functions of the form are catenaries. (Figure) shows the graph of

### Using a Catenary to Find the Length of a Cable

Assume a hanging cable has the shape for where is measured in feet. Determine the length of the cable (in feet).

#### Solution

Recall from Section 6.4 that the formula for arc length is

We have so Then

Now recall that so we have

Assume a hanging cable has the shape for Determine the length of the cable (in feet).

#### Hint

Use the procedure from the previous example.

### Key Concepts

• Hyperbolic functions are defined in terms of exponential functions.
• Term-by-term differentiation yields differentiation formulas for the hyperbolic functions. These differentiation formulas give rise, in turn, to integration formulas.
• With appropriate range restrictions, the hyperbolic functions all have inverses.
• Implicit differentiation yields differentiation formulas for the inverse hyperbolic functions, which in turn give rise to integration formulas.
• The most common physical applications of hyperbolic functions are calculations involving catenaries.

1. [T] Find expressions for and Use a calculator to graph these functions and ensure your expression is correct.

#### Solution

2. From the definitions of and find their antiderivatives.

3. Show that and satisfy

#### Solution

4. Use the quotient rule to verify that

5. Derive from the definition.

#### Solution

6. Take the derivative of the previous expression to find an expression for

7. Prove by changing the expression to exponentials.

#### Solution

8. Take the derivative of the previous expression to find an expression for

For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct.

9. [T]

10. [T]

11. [T]

12. [T]

13. [T]

14. [T]

15. [T]

16. [T]

17. [T]

#### Solution

18. [T]

For the following exercises, find the antiderivatives for the given functions.

19.

20.

21.

23.

24.

25.

26.

27.

28.

#### Solution

29.

For the following exercises, find the derivatives for the functions.

30.

31.

32.

33.

34.

#### Solution

35.

36.

For the following exercises, find the antiderivatives for the functions.

37.

38.

39.

40.

41.

42.

#### Solution

43.

For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation

44. Show that satisfies this equation.

#### Solution

45. Derive the previous expression for by integrating

46. [T] Estimate how far a body has fallen in 12 seconds by finding the area underneath the curve of

47.

#### Solution

37.30

For the following exercises, use this scenario: A cable hanging under its own weight has a slope that satisfies The constant is the ratio of cable density to tension.

48. Show that satisfies this equation.

49. Integrate to find the cable height if

#### Solution

50. Sketch the cable and determine how far down it sags at

For the following exercises, solve each problem.

51. [T] A chain hangs from two posts 2 m apart to form a catenary described by the equation Find the slope of the catenary at the left fence post.

#### Solution

-0.521095

52. [T] A chain hangs from two posts four meters apart to form a catenary described by the equation Find the total length of the catenary (arc length).

53. [T] A high-voltage power line is a catenary described by Find the ratio of the area under the catenary to its arc length. What do you notice?

#### Solution

10

54. A telephone line is a catenary described by Find the ratio of the area under the catenary to its arc length. Does this confirm your answer for the previous question?

55. Prove the formula for the derivative of by differentiating (Hint: Use hyperbolic trigonometric identities.)

56. Prove the formula for the derivative of by differentiating

(Hint: Use hyperbolic trigonometric identities.)

57. Prove the formula for the derivative of by differentiating (Hint: Use hyperbolic trigonometric identities.)

58. Prove that

59. Prove the expression for Multiply by and solve for Does your expression match the textbook?

60. Prove the expression for Multiply by and solve for Does your expression match the textbook?

## Glossary

catenary
a curve in the shape of the function is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary