The exponential function is perhaps the most efficient function in terms of the operations of calculus. The exponential function, [latex]y={e}^{x},[/latex] is its own derivative and its own integral.
Integrals of Exponential Functions
Exponential functions can be integrated using the following formulas.
The nature of the antiderivative of [latex]{e}^{x}[/latex] makes it fairly easy to identify what to choose as [latex]u[/latex]. If only one [latex]e[/latex] exists, choose the exponent of [latex]e[/latex] as [latex]u[/latex]. If more than one [latex]e[/latex] exists, choose the more complicated function involving [latex]e[/latex] as [latex]u[/latex].
Example: Finding an Antiderivative of an Exponential Function
Find the antiderivative of the exponential function [latex]e^{-x}[/latex].
Show Solution
Use substitution, setting [latex]u=\text{−}x,[/latex] and then [latex]du=-1dx.[/latex] Multiply the du equation by −1, so you now have [latex]\text{−}du=dx.[/latex] Then,
Watch the following video to see the worked solution to the above Try It.
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A common mistake when dealing with exponential expressions is treating the exponent on [latex]e[/latex] the same way we treat exponents in polynomial expressions. We cannot use the power rule for the exponent on [latex]e[/latex]. 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.
Example: Square Root of an Exponential Function
Find the antiderivative of the exponential function [latex]{e}^{x}\sqrt{1+{e}^{x}}.[/latex]
Show Solution
First rewrite the problem using a rational exponent:
Watch the following video to see the worked solution to the above Try It.
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Example: Using Substitution with an Exponential Function
Use substitution to evaluate the indefinite integral [latex]\displaystyle\int 3{x}^{2}{e}^{2{x}^{3}}dx.[/latex]
Show Solution
Here we choose to let [latex]u[/latex] equal the expression in the exponent on [latex]e[/latex]. Let [latex]u=2{x}^{3}[/latex] and [latex]du=6{x}^{2}dx..[/latex] Again, du is off by a constant multiplier; the original function contains a factor of 3[latex]x[/latex]2, not 6[latex]x[/latex]2. Multiply both sides of the equation by [latex]\frac{1}{2}[/latex] so that the integrand in [latex]u[/latex] equals the integrand in [latex]x[/latex]. Thus,
As mentioned at the beginning of this section, exponential functions are used in many real-life applications. The number e 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.
Example: 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, [latex]{p}^{\prime }(x),[/latex] for [latex]x[/latex] number of tubes per week, is given as
[latex]p\text{‘}(x)=-0.015{e}^{-0.01x}.[/latex]
If the supermarket chain sells 100 tubes per week, what price should it set?
Show Solution
To find the price–demand equation, integrate the marginal price–demand function. First find the antiderivative, then look at the particulars. Thus,
The supermarket should charge $1.99 per tube if it is selling 100 tubes per week.
Watch the following video to see the worked solution to Example: Finding a Price–Demand Equation.
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example: Evaluating a Definite Integral Involving an Exponential Function
Evaluate the definite integral [latex]{\displaystyle\int }_{1}^{2}{e}^{1-x}dx.[/latex]
Show Solution
Again, substitution is the method to use. Let [latex]u=1-x,[/latex] so [latex]du=-1dx[/latex] or [latex]\text{−}du=dx.[/latex] Then [latex]\displaystyle\int {e}^{1-x}dx=\text{−}\displaystyle\int {e}^{u}du.[/latex] Next, change the limits of integration. Using the equation [latex]u=1-x,[/latex] we have
Suppose the rate of growth of bacteria in a Petri dish is given by [latex]q(t)={3}^{t},[/latex] where [latex]t[/latex] is given in hours and [latex]q(t)[/latex] is given in thousands of bacteria per hour. If a culture starts with 10,000 bacteria, find a function [latex]Q(t)[/latex] that gives the number of bacteria in the Petri dish at any time [latex]t[/latex]. How many bacteria are in the dish after 2 hours?
After 2 hours, there are 17,282 bacteria in the dish.
Try It
From (Figure), suppose the bacteria grow at a rate of [latex]q(t)={2}^{t}.[/latex] Assume the culture still starts with 10,000 bacteria. Find [latex]Q(t).[/latex] How many bacteria are in the dish after 3 hours?
Hint
Use the procedure from (Figure) to solve the problem.
Show Solution
[latex]Q(t)=\frac{{2}^{t}}{\text{ln}2}+8.557.[/latex] There are 20,099 bacteria in the dish after 3 hours.
Fruit Fly Population Growth
Suppose a population of fruit flies increases at a rate of [latex]g(t)=2{e}^{0.02t},[/latex] in flies per day. If the initial population of fruit flies is 100 flies, how many flies are in the population after 10 days?
Show Solution
Let [latex]G(t)[/latex] represent the number of flies in the population at time [latex]t[/latex]. Applying the net change theorem, we have
There are 122 flies in the population after 10 days.
Try It
Suppose the rate of growth of the fly population is given by [latex]g(t)={e}^{0.01t},[/latex] and the initial fly population is 100 flies. How many flies are in the population after 15 days?
Hint
Use the process from (Figure) to solve the problem.
Show Solution
There are 116 flies.
example: Evaluating a Definite Integral Using Substitution
Evaluate the definite integral using substitution: [latex]{\displaystyle\int }_{1}^{2}\dfrac{{e}^{1\text{/}x}}{{x}^{2}}dx.[/latex]
Show Solution
This problem requires some rewriting to simplify applying the properties. First, rewrite the exponent on [latex]e[/latex] as a power of [latex]x[/latex], then bring the [latex]x[/latex]2 in the denominator up to the numerator using a negative exponent. We have
![A graph of the function f(x) = e^x * sqrt(1 + e^x), which is an increasing concave up curve, over [-3, 1]. It begins close to the x-axis in quadrant two, crosses the y-axis at (0, sqrt(2)), and continues to increase rapidly.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/2332/2018/01/11204252/CNX_Calc_Figure_05_06_001.jpg)
![A graph of the function f(x) = e^(1-x) over [0, 3]. It crosses the y-axis at (0, e) as a decreasing concave up curve and symptotically approaches 0 as x goes to infinity.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/2332/2018/01/11204254/CNX_Calc_Figure_05_06_002.jpg)
