In this explainer, we will learn how to find the indefinite integral of exponential and reciprocal functions .
Indefinite integrals of exponential and logarithmic functions have many real-world applications as the functions are used in mathematical models to describe population growth, cell growth, and radioactive decay.
These types of problems can be solved by using the following rules.
Definition: Integrals of Exponential and Reciprocal Functions
for to avoid dividing by zero in the first expression.
We can verify these directly using the first part of the fundamental theorem of calculus.
Definition: The Fundamental Theorem of Calculus (FTC)
Let be a continuous real-valued function defined on . Let be the function defined, for all in , by
Then, is uniformly continuous on and differentiable on , and for all in .
By taking the derivative of the right-hand side for each of the statements above, we can show the result equals the integrand:
Therefore, upon integrating these, we have
Now, what if we want to integrate something like (i.e., an exponential expression with an arbitrary base)? The trick is to use the fact that and apply the laws of logarithms with :
We can use the standard rule for exponentials (1) with base to obtain where to avoid dividing by zero. More generally, we have for and . We do not need to memorize this result as we can derive it directly from applying the laws of logarithms for each exponential base.
Let’s look at an integral with both exponential and reciprocal functions given by
We know that the integral of the sum or difference of two functions is equal to the sum or difference of the integrals of those functions. In other words, we can separate out the two parts and take constant factors outside the integral. For each part, we also get a constant of integration, which we can combine into a single constant .
Applying the standard rules for integrating exponential (1) and reciprocal (2) functions as given in the definition, we find
The constant of integration can also be determined if there is a boundary condition. Suppose is a boundary condition. We can substitute this in order to determine the constant as which after rearranging gives
Therefore, can be written as
Now let’s look at a few examples to practice and deepen our understanding. The first two examples contain exponential functions of different bases.
Example 1: Finding the Integration of a Function Containing Exponential Functions by Distributing the Division
In this example, we want to find the indefinite integral of a function containing exponentials with base .
We begin by separating each part of the numerator and dividing individually by in the integrand to give us
Next, separating the integral and taking any constant factors outside, we can use the standard rule for integrating exponential terms:
We obtain a constant of integration for these separate parts, but we can combine them into a single constant . In particular, we have
In our first example, we saw how to integrate a number of exponential functions with base . We will now demonstrate how we can use the laws of exponentials and logarithms to integrate an exponential function with a base of 2.
Example 2: Finding the Integration of an Exponential Function with an Integer Base
In this example, we want to find the indefinite integral of a function containing an exponential with base 2.
We first begin by rewriting the integrand in base as where we have applied the laws of logarithms to obtain the final form. We can now use the standard rule for integrating exponential terms:
In particular, we find
In our next example, we will demonstrate how to apply the rule (2), given in the definition, to integrate reciprocal functions.
Example 3: Finding the Integration of a Reciprocal Function
In this example, we want to find the indefinite integral of a reciprocal function.
This integral can be performed simply by taking the constant outside the integral and applying the standard rule for the integral reciprocal functions:
In particular, we have
Now, let’s look at an example where we have to apply the same rules to an integral after distributing the division.
Example 4: Finding the Integration of a Rational Function by Distributing the Division
In this example, we want to find the indefinite integral of a rational function. Often, our instinct in these cases might be to apply integration by substitution. However, we notice that if we distribute the parentheses, we can simplify the integrand by dividing each term of the numerator by the in the denominator.
To begin, we distribute the parentheses in the integrand to obtain a quadratic expression in the numerator and then perform the division by :
The first expression in the integrand contains a linear term, which can be integrated using the power rule, and the second expression is a reciprocal, which can be integrated using the standard rule for integrating reciprocals:
In particular, we find
Until this stage, we have applied our rules to find the general solution to integrals involving exponentials and logarithms. We can find a particular solution to these sorts of problems by applying a boundary condition to help us determine the constant of integration.
Example 5: Applying the Fundamental Theorem of Calculus to Integrate a Rational Function with Boundary Conditions
Find, if possible, an antiderivative of that satisfies the conditions and .
In this example, we want to find the antiderivative of a reciprocal function satisfying particular boundary conditions.
We begin by using the first part of the fundamental theorem of calculus to determine ; it is the indefinite integral of :
We might recognize the result, or we can use integration by substitution. For the latter, we note that the integrand contains a composite function: with and . Thus, we can use the substitution:
The derivative of this with respect to is or, equivalently, by manipulating the differentials,
Next, we apply this substitution to the integral to change the variable from to and integrate the resulting expression using the standard rule for integrating reciprocal functions given by
Thus, we obtain
Now, we apply the reverse substitution to get everything back in terms of :
Now, we can use the boundary conditions and . Since our solution to the integral involves a modulus, we recall the definition of :
We also note that the natural , , is undefined at ; thus, for , we have to consider values of :
We can also split up the constant of integration , in each part of the piecewise function, for some as
This is essential since the constant will be different, as we shall see, depending on the value of where the function is defined. Therefore,
The boundary condition can only be applied to the second part with as . This allows us to determine as hence,
Similarly, the boundary condition can only be applied to the first part with as . This allows us to determine : hence,
Therefore, the antiderivative of , which satisfies the given boundary conditions, is given by
In our next example, we will look at how to apply the rules for integrating reciprocal functions involving a root and negative exponents.
Example 6: Finding the Integration of a Function Involving Expanding Squares and Using the Power Rule with Negative Exponents
In this example, we want to find the indefinite integral of a function involving and expanding squares.
We begin by distributing the square in the integrand:
Now, we can separate the integrals into two parts: the first expression containing a linear term that can be integrated using the power rule and the second expression a reciprocal term that can be integrated using the standard rule:
In particular, we find
In our final example, we will apply the integration of exponentials twice in order to determine a function from its second derivative.
Example 7: Finding the Expression of a Function given Its Second Derivative Using Indefinite Integration
Given that , find .
In this example, we want to find the function from the expression we have for its second derivative. Since integration is the reverse process of differentiation, we can determine by performing two successive integrals.
We begin first by finding by using the first part of the fundamental theorem of calculus. Integrating the expression for the second derivative using the general power rule and the standard rule for exponentials, we obtain
Finally, we determine by using the fundamental theorem of calculus again and performing another integral with the expression of the first derivative:
- Indefinite integrals involving exponentials and reciprocals can be evaluated using the standard results:
- To find the integral of an exponential term with an arbitrary base, we can apply the substitution to derive the following general result: and