5. Integration

# 5.6 Integrals Involving Exponential and Logarithmic Functions

### Learning Objectives

• Integrate functions involving exponential functions.
• Integrate functions involving logarithmic functions.

Exponential and logarithmic functions are used to model population growth, cell growth, and financial growth, as well as depreciation, radioactive decay, and resource consumption, to name only a few applications. In this section, we explore integration involving exponential and logarithmic functions.

# Integrals of Exponential Functions

The exponential function is perhaps the most efficient function in terms of the operations of calculus. The exponential function, is its own derivative and its own integral.

### Rule: Integrals of Exponential Functions

Exponential functions can be integrated using the following formulas.

### Finding an Antiderivative of an Exponential Function

Find the antiderivative of the exponential function .

#### Solution

Use substitution, setting and then Multiply the du equation by −1, so you now have Then,

Find the antiderivative of the function using substitution:

#### Hint

Let equal the exponent on .

A common mistake when dealing with exponential expressions is treating the exponent on the same way we treat exponents in polynomial expressions. We cannot use the power rule for the exponent on . This can be especially confusing when we have both exponentials and polynomials in the same expression, as in the previous checkpoint. In these cases, we should always double-check to make sure we’re using the right rules for the functions we’re integrating.

### Square Root of an Exponential Function

Find the antiderivative of the exponential function

#### Solution

First rewrite the problem using a rational exponent:

Using substitution, choose Then, We have ((Figure))

Then

Find the antiderivative of

Let

### Using Substitution with an Exponential Function

Use substitution to evaluate the indefinite integral

#### Solution

Here we choose to let equal the expression in the exponent on . Let and Again, du is off by a constant multiplier; the original function contains a factor of 32, not 62. Multiply both sides of the equation by so that the integrand in equals the integrand in . Thus,

Integrate the expression in and then substitute the original expression in back into the integral:

Evaluate the indefinite integral

#### Hint

Let

As mentioned at the beginning of this section, exponential functions are used in many real-life applications. The number is often associated with compounded or accelerating growth, as we have seen in earlier sections about the derivative. Although the derivative represents a rate of change or a growth rate, the integral represents the total change or the total growth. Let’s look at an example in which integration of an exponential function solves a common business application.

A price–demand function tells us the relationship between the quantity of a product demanded and the price of the product. In general, price decreases as quantity demanded increases. The marginal price–demand function is the derivative of the price–demand function and it tells us how fast the price changes at a given level of production. These functions are used in business to determine the price–elasticity of demand, and to help companies determine whether changing production levels would be profitable.

### Finding a Price–Demand Equation

Find the price–demand equation for a particular brand of toothpaste at a supermarket chain when the demand is 50 tubes per week at $2.35 per tube, given that the marginal price—demand function, for number of tubes per week, is given as If the supermarket chain sells 100 tubes per week, what price should it set? #### Solution To find the price–demand equation, integrate the marginal price–demand function. First find the antiderivative, then look at the particulars. Thus, Using substitution, let and Then, divide both sides of the du equation by −0.01. This gives The next step is to solve for C. We know that when the price is$2.35 per tube, the demand is 50 tubes per week. This means

Now, just solve for C:

Thus,

If the supermarket sells 100 tubes of toothpaste per week, the price would be

The supermarket should charge \$1.99 per tube if it is selling 100 tubes per week.

### Evaluating a Definite Integral Involving an Exponential Function

Evaluate the definite integral

#### Solution

Again, substitution is the method to use. Let so or Then Next, change the limits of integration. Using the equation we have

The integral then becomes

See (Figure).

Evaluate

Let

### Growth of Bacteria in a Culture

Suppose the rate of growth of bacteria in a Petri dish is given by where is given in hours and is given in thousands of bacteria per hour. If a culture starts with 10,000 bacteria, find a function that gives the number of bacteria in the Petri dish at any time . How many bacteria are in the dish after 2 hours?

#### Solution

We have

Then, at we have so and we get

At time we have

After 2 hours, there are 17,282 bacteria in the dish.

From (Figure), suppose the bacteria grow at a rate of Assume the culture still starts with 10,000 bacteria. Find How many bacteria are in the dish after 3 hours?

#### Solution

There are 20,099 bacteria in the dish after 3 hours.

#### Hint

Use the procedure from (Figure) to solve the problem.

### Fruit Fly Population Growth

Suppose a population of fruit flies increases at a rate of in flies per day. If the initial population of fruit flies is 100 flies, how many flies are in the population after 10 days?

#### Solution

Let represent the number of flies in the population at time . Applying the net change theorem, we have

There are 122 flies in the population after 10 days.

Suppose the rate of growth of the fly population is given by and the initial fly population is 100 flies. How many flies are in the population after 15 days?

#### Solution

There are 116 flies.

#### Hint

Use the process from (Figure) to solve the problem.

### Evaluating a Definite Integral Using Substitution

Evaluate the definite integral using substitution:

#### Solution

This problem requires some rewriting to simplify applying the properties. First, rewrite the exponent on as a power of , then bring the 2 in the denominator up to the numerator using a negative exponent. We have

Let the exponent on . Then

Bringing the negative sign outside the integral sign, the problem now reads

Next, change the limits of integration:

Notice that now the limits begin with the larger number, meaning we must multiply by −1 and interchange the limits. Thus,

Evaluate the definite integral using substitution:

Let

# Integrals Involving Logarithmic Functions

Integrating functions of the form result in the absolute value of the natural log function, as shown in the following rule. Integral formulas for other logarithmic functions, such as and are also included in the rule.

### Rule: Integration Formulas Involving Logarithmic Functions

The following formulas can be used to evaluate integrals involving logarithmic functions.

### Finding an Antiderivative Involving

Find the antiderivative of the function

#### Solution

First factor the 3 outside the integral symbol. Then use the −1 rule. Thus,

See (Figure).

Find the antiderivative of

#### Hint

Follow the pattern from (Figure) to solve the problem.

### Finding an Antiderivative of a Rational Function

Find the antiderivative of

This can be rewritten as Use substitution. Let then Alter du by factoring out the 2. Thus,

Rewrite the integrand in :

Then we have

### Finding an Antiderivative of a Logarithmic Function

Find the antiderivative of the log function

#### Solution

Follow the format in the formula listed in the rule on integration formulas involving logarithmic functions. Based on this format, we have

Find the antiderivative of

#### Hint

Follow (Figure) and refer to the rule on integration formulas involving logarithmic functions.

(Figure) is a definite integral of a trigonometric function. With trigonometric functions, we often have to apply a trigonometric property or an identity before we can move forward. Finding the right form of the integrand is usually the key to a smooth integration.

### Evaluating a Definite Integral

Find the definite integral of

#### Solution

We need substitution to evaluate this problem. Let so Rewrite the integral in terms of , changing the limits of integration as well. Thus,

Then

### Key Concepts

• Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay.
• Substitution is often used to evaluate integrals involving exponential functions or logarithms.

# Key Equations

• Integrals of Exponential Functions

• Integration Formulas Involving Logarithmic Functions

In the following exercises, compute each indefinite integral.

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#### Solution

In the following exercises, find each indefinite integral by using appropriate substitutions.

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#### Solution

In the following exercises, verify by differentiation that then use appropriate changes of variables to compute the integral.

23. $(Hint\text{:}\int \text{ln}xdx=\frac{1}{2}\int x\text{ln}({x}^{2})dx)$

24.

#### Solution

25. $(Hint\text{:}\text{Set}u=\frac{1}{x}\text{.})$

26. $(Hint\text{:}\text{Set}u=\sqrt{x}\text{.})$

#### Solution

27. Write an integral to express the area under the graph of from to ex and evaluate the integral.

28. Write an integral to express the area under the graph of between and and evaluate the integral.

#### Solution

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms.

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#### Solution

35.

In the following exercises, evaluate the definite integral.

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#### Solution

In the following exercises, integrate using the indicated substitution.

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#### Solution

In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R50 and solve for the exact area.

47. [T] over

48. [T] over

#### Solution

Exact solution: Since is decreasing, the right endpoint estimate underestimates the area.

49. [T] over

50. [T] over

#### Solution

Exact solution: Since is increasing, the right endpoint estimate overestimates the area.

51. [T] over

52. [T] over

#### Solution

Exact solution: Since is increasing, the right endpoint estimate overestimates the area (the actual area is a larger negative number).

In the following exercises, for Find the area under the graph of between the given values and by integrating.

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#### Solution

57. Find the area under the graph of the function between and

58. Compute the integral of and find the smallest value of N such that the area under the graph between and is, at most, 0.01.

#### Solution

The quantity is less than 0.01 when

59. Find the limit, as N tends to infinity, of the area under the graph of between and

60. Show that when

#### Solution

61. Suppose that for all and that and are differentiable. Use the identity and the chain rule to find the derivative of

62. Use the previous exercise to find the antiderivative of and evaluate

#### Solution

23

63. Show that if then the integral of from ac to bc is the same as the integral of from to .

The following exercises are intended to derive the fundamental properties of the natural log starting from the Definition/em> using properties of the definite integral and making no further assumptions.

64. Use the identity to derive the identity

#### Solution

We may assume that Then, Now make the substitution so and and change endpoints:

65. Use a change of variable in the integral to show that

66. Use the identity to show that is an increasing function of on and use the previous exercises to show that the range of is Without any further assumptions, conclude that has an inverse function defined on

67. Pretend, for the moment, that we do not know that is the inverse function of but keep in mind that has an inverse function defined on Call it E. Use the identity to deduce that for any real numbers , .

68. Pretend, for the moment, that we do not know that is the inverse function of but keep in mind that has an inverse function defined on Call it E. Show that

#### Solution

Then, Since any number can be written for some , and for such we have it follows that for any

69. The sine integral, defined as is an important quantity in engineering. Although it does not have a simple closed formula, it is possible to estimate its behavior for large . Show that for (Hint: )

70. [T] The normal distribution in probability is given by where σ is the standard deviation and μ is the average. The standard normal distribution in probability, corresponds to Compute the left endpoint estimates of

#### Solution

71. [T] Compute the right endpoint estimates of