## Integrals of Exponential Functions

### Learning Outcomes

• Integrate functions involving exponential functions

The exponential function is perhaps the most efficient function in terms of the operations of calculus. The exponential function, $y={e}^{x},$ is its own derivative and its own integral.

### Integrals of Exponential Functions

Exponential functions can be integrated using the following formulas.

$\begin{array}{ccc} {\displaystyle\int{e}^{x}dx} & {=} & {{e}^{x}+C} \\ {\displaystyle\int{a}^{x}dx} & {=} & {\dfrac{{a}^{x}}{\text{ln}a}+C}\end{array}$

The nature of the antiderivative of ${e}^{x}$ makes it fairly easy to identify what to choose as $u$. If only one $e$ exists, choose the exponent of $e$ as $u$. If more than one $e$ exists, choose the more complicated function involving $e$ as $u$.

### Example: Finding an Antiderivative of an Exponential Function

Find the antiderivative of the exponential function $e^{-x}$.

### Try It

Find the antiderivative of the function using substitution: ${x}^{2}{e}^{-2{x}^{3}}.$

Watch the following video to see the worked solution to the above Try It.

A common mistake when dealing with exponential expressions is treating the exponent on $e$ the same way we treat exponents in polynomial expressions. We cannot use the power rule for the exponent on $e$. 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 ${e}^{x}\sqrt{1+{e}^{x}}.$

### Try It

Find the antiderivative of ${e}^{x}{(3{e}^{x}-2)}^{2}.$

Watch the following video to see the worked solution to the above Try It.

### Example: Using Substitution with an Exponential Function

Use substitution to evaluate the indefinite integral $\displaystyle\int 3{x}^{2}{e}^{2{x}^{3}}dx.$

### Try It

Evaluate the indefinite integral $\displaystyle\int 2{x}^{3}{e}^{{x}^{4}}dx.$

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.

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, ${p}^{\prime }(x),$ for $x$ number of tubes per week, is given as

$p\text{‘}(x)=-0.015{e}^{-0.01x}.$

If the supermarket chain sells 100 tubes per week, what price should it set?

Watch the following video to see the worked solution to Example: Finding a Price–Demand Equation.

### example: Evaluating a Definite Integral Involving an Exponential Function

Evaluate the definite integral ${\displaystyle\int }_{1}^{2}{e}^{1-x}dx.$

### Try It

Evaluate ${\displaystyle\int }_{0}^{2}{e}^{2x}dx.$

### Example: Growth of Bacteria in a Culture

Suppose the rate of growth of bacteria in a Petri dish is given by $q(t)={3}^{t},$ where $t$ is given in hours and $q(t)$ is given in thousands of bacteria per hour. If a culture starts with 10,000 bacteria, find a function $Q(t)$ that gives the number of bacteria in the Petri dish at any time $t$. How many bacteria are in the dish after 2 hours?

### Try It

From (Figure), suppose the bacteria grow at a rate of $q(t)={2}^{t}.$ Assume the culture still starts with 10,000 bacteria. Find $Q(t).$ How many bacteria are in the dish after 3 hours?

### Fruit Fly Population Growth

Suppose a population of fruit flies increases at a rate of $g(t)=2{e}^{0.02t},$ in flies per day. If the initial population of fruit flies is 100 flies, how many flies are in the population after 10 days?

### Try It

Suppose the rate of growth of the fly population is given by $g(t)={e}^{0.01t},$ and the initial fly population is 100 flies. How many flies are in the population after 15 days?

### example: Evaluating a Definite Integral Using Substitution

Evaluate the definite integral using substitution: ${\displaystyle\int }_{1}^{2}\dfrac{{e}^{1\text{/}x}}{{x}^{2}}dx.$

### Try It

Evaluate the definite integral using substitution: ${\displaystyle\int }_{1}^{2}\frac{1}{{x}^{3}}{e}^{4{x}^{-2}}dx.$