Integrals of Exponential Functions

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TABLE OF CONTENTS

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Finding the derivative of an exponential function is pretty straightforward since its derivative is the exponential function itself, so we might be tempted to assume that finding the integrals of exponential functions is not a big deal.

This is not the case at all. Differentiation is a straightforward operation, while integration is not. Even if we want to integrate an exponential function, we must pay special attention to the integrand and use an appropriate integration technique.

Integrals of Exponential Functions

We begin by recalling how to differentiate an exponential function.

The derivative of the natural exponential function is the natural exponential function itself.

$$\dfrac{\mathrm{d}}{\mathrm{d}x}e^x=e^x$$

If the base is other than $e$, then we need to multiply by the natural logarithm of the base.

$$\dfrac{\mathrm{d}}{\mathrm{d}x}a^x=\ln{a}\, a^x$$

Of course, we also have to use any differentiation rules as needed! Let's take a look at a quick example using The Chain Rule.

Find the derivative of $f (x) = e^{2 x^{2}}$ .

Let $u = 2 x^{2}$ and differentiate using The Chain Rule.

$\frac{d f}{d x} = \frac{d}{d u} e^{u} \frac{d u}{d x}$

Differentiate the exponential function.

$\frac{d f}{d x} = e^{u} \frac{d u}{d x}$

Use The Power Rule to differentiate $u = 2 x^{2}$ .

$\frac{d u}{d x} = 4 x$

Substitute back $u = 2 x^{2}$ and $\frac{d u}{d x} = 4 x$ .

$\frac{d f}{d x} = e^{2 x^{2}} 4 x$

Rearrange the expression.

$\frac{d f}{d x} = 4 x e^{2 x^{2}}$

We will now take a look at how to integrate exponential functions. The derivative of the exponential function is the exponential function itself, so we can also think of this as if the exponential function is its own antiderivative.

The antiderivative of the exponential function is the exponential function itself.

$\int e^{x} d x = e^{x} + C$

If the base is other than $e$ you divide by the natural logarithm of the base.

$$\int a^x\mathrm{d}x=\dfrac{1}{\ln{a}}a^x+C$$

Do not forget to add +C when finding the antiderivative of functions!

Let's see a quick example of the integral of an exponential function.

Evaluate the integral $\int e^{3 x} d x$ .

Since the argument of the exponential function is 3x, we need to do Integration by Substitution.

Let $u = 3 x$ . Find du using The Power Rule.

$u = 3 x \to \frac{d u}{d x} = 3$

$\frac{d u}{d x} = 3 \to d u = 3 d x$

Isolate dx.

$d x = \frac{1}{3} d u$

Substitute $u = 3 x$ and $d x = \frac{1}{3} d u$ in the integral.

$\int e^{3 x} d x = \int e^{u} \frac{1}{3} d u$

Rearrange the integral.

$\int e^{3 x} = \frac{1}{3} \int e^{u} d u$

Integrate the exponential function.

$\int e^{3 x} d x = \frac{1}{3} e^{u} + C$

Substitute back $u = 3 x$ in the integral.

$\int e^{3 x} d x = \frac{1}{3} e^{3 x} + C$

Be sure to use any of the Integration Techniques as needed!

We can avoid using Integration by Substitution if the argument of the exponential function is a multiple of x.

If the argument of the exponential function is a multiple of x, then its antiderivative is the following:

$\int e^{a x} d x = \frac{1}{a} e^{a x} + C$

Where $a$ is any real number constant other than 0.

The above formula will make our lives easier when integrating exponential functions!

Definite Integrals of Exponential Functions

How about the evaluation of definite integrals that involve exponential functions? No problem! We can use The Fundamental Theorem of Calculus to do so!

Evaluate the definite integral $\int_{0}^{1} e^{x} d x$ .

Find the antiderivative of $e^{x}$ .

$\int e^{x} = e^{x} + C$

Use The Fundamental Theorem of Calculus to evaluate the definite integral.

$\int_{0}^{1} e^{x} d x = {(e^{x} + C)}_{0}^{1}$

$\int_{0}^{1} e^{x} d x = (e^{1} + C) - (e^{0} + C)$

Use the properties of exponents and simplify.

$\int_{0}^{1} e^{x} d x = e - 1$

Up to this point, we have an exact result. You can always use a calculator if you need to know the integral's numerical value.

Use a calculator to find the numerical value of the definite integral.

$\int_{0}^{1} e^{x} d x = 1.718281828 . . .$

We can also evaluate improper integrals knowing the following limits of the exponential function.

The limit of the exponential function as x tends to negative infinity is equal to 0. This can be expressed in two ways with the following formulas.

$\lim_{x \to - \infty} e^{x} = 0$

$\lim_{x \to \infty} e^{- x} = 0$

These limits will allow us to evaluate improper integrals involving exponential functions. This is better understood with an example. Let's do it!

Evaluate the definite integral $\int_{0}^{\infty} e^{- 2 x} d x$ .

Begin by finding the antiderivative of the given function.

Let $u = - 2 x$ . Find du using The Power Rule.

$u = - 2 x \to \frac{d u}{d x} = - 2$

$\frac{d u}{d x} = - 2 \to d u = - 2 d x$

Isolate $d x$ .

$d x = - \frac{1}{2} d u$

Substitute $u = - 2 x$ and $d x = - \frac{1}{2} d u$ in the integral.

$\int e^{- 2 x} d x = \int e^{u} (- \frac{1}{2} d u)$

Rearrange the integral.

$\int e^{- 2 x} d x = - \frac{1}{2} \int e^{u} d u$

Integrate the exponential function.

$\int e^{- 2 x} d x = - \frac{1}{2} e^{u} + C$

Substitute back $u = - 2 x$ .

$\int e^{- 2 x} d x = - \frac{1}{2} e^{- 2 x} + C$

In order to evaluate the improper integral, we use The Fundamental Theorem of Calculus, but we evaluate the upper limit as it goes to infinity. That is, we let $b\rightarrow\infty$ in the upper integration limit.

$\int_{0}^{\infty} e^{- 2 x} d x = \lim_{b \to \infty} (- \frac{1}{2} e^{- 2 b} + C) - (- \frac{1}{2} e^{- 2 (0)} + C)$

Simplify using the Properties of Limits.

$\int_{0}^{\infty} e^{- 2 x} d x = - \frac{1}{2} (\lim_{b \to \infty} e^{- 2 b} - e^{0})$

As $b$ goes to infinity, the argument of the exponential function goes to negative infinity, so we can use the following limit:

$\lim_{x \to \infty} e^{- x} = 0$

We also note that $e^{0} = 1$ . Knowing this, we can find the value of our integral.

Evaluate the limit as $b \to \infty$ and substitute $e^{0} = 1$ .

$\int_{0}^{\infty} e^{- 2 x} d x = - \frac{1}{2} (0 - 1)$

Simplify.

$\int_{0}^{\infty} e^{- 2 x} d x = \frac{1}{2}$

Integrals of Exponential Functions Examples

Integrating is kind of a special operation in calculus. We need to have insight on which integration technique is to be used. How do we get better at integrating? With practice, of course! Let's see more examples of integrals of exponential functions!

Evaluate the integral $\int 2 x e^{x^{2}} d x$ .

Note that this integral involves $x^{2}$ and $2 x$ in the integrand. Since these two expressions are related by a derivative, we will do Integration by Substitution.

Let $u = x^{2}$ . Find $d u$ using The Power Rule.

$u = x^{2} \to \frac{d u}{d x} = 2 x$

$\frac{d u}{d x} = 2 x \to d u = 2 x d x$

Rearrange the integral.

$\int 2 x e^{x^{2}} d x = \int e^{x^{2}} (2 x d x)$

Substitute $u = x^{2}$ and $d u = 2 x d x$ in the integral.

$\int 2 x e^{x^{2}} d x = \int e^{u} d u$

Integrate the exponential function.

$\int 2 x e^{x^{2}} d x = e^{u} + C$

Substitute back $u = x^{2}$ .

$\int 2 x e^{x^{2}} d x = e^{x^{2}} + C$

Sometimes we will need to use Integration by Parts several times! Need a refresher on the topic? Take a look at our Integration by Parts article!

Evaluate the integral $\int (x^{2} + 3 x) e^{x} d x$

Use LIATE to make an appropriate choice of u and dv.

$u = x^{2} + 3 x$

$d v = e^{x} d x$

Use The Power Rule to find du.

$d u = (2 x + 3) d x$

Integrate the exponential function to find v.

$v = \int e^{x} d x = e^{x}$

Use the Integration by Parts formula $\int u d v = u v - \int v d u$

$\int (x^{2} + 3 x) e^{x} d x = (x^{2} + 3 x) e^{x} - \int e^{x} (2 x + 3) d x$

The resulting integral on the right-hand side of the equation can also be done by Integration by Parts. We shall focus on evaluating $\int e^{x} (2 x + 3) d x$ to avoid any confusion.

Use LIATE to make an appropriate choice of u and dv.

$u = 2 x + 3$

$d v = e^{x} d x$

Use The Power Rule to find du.

$d u = 2 d x$

Integrate the exponential function to find v.

$v = \int e^{x} d x = e^{x}$

Use the Integration by Parts formula.

$\int e^{x} (2 x + 3) d x = (2 x + 3) e^{x} - \int e^{x} (2 d x)$

Integrate the exponential function.

$\int e^{x} (2 x + 3) d x = (2 x + 3) e^{x} - 2 e^{x}$

Substitute back the above integral into the original integral and add the integration constant C.

$\int (x^{2} + 3 x) e^{x} d x = (x^{2} + 3 x) e^{x} - [(2 x + 3) e^{x} - 2 e^{x}] + C$

Simplify by factoring out $e^{x}$ .

$\int (x^{2} + 3 x) e^{3 x} d x = e^{x} (x^{2} + x - 1) + C$

Let's see one more example involving a definite integral.

Evaluate the integral $\int_{1}^{2} e^{- 4 x} d x$ .

Begin by finding the antiderivative of the function. Then we can evaluate the definite integral using The Fundamental Theorem of Calculus.

Integrate the exponential function.

$\int e^{- 4 x} d x = - \frac{1}{4} e^{- 4 x} + C$

Use The Fundamental Theorem of Calculus to evaluate the definite integral.

$\int_{1}^{2} e^{- 4 x} d x = {(- \frac{1}{4} e^{- 4 x} + C)}_{1}^{2}$

$\int_{1}^{2} e^{- 4 x} d x = (- \frac{1}{4} e^{- 4 (2)} + C) - (- \frac{1}{4} e^{- 4 (1)} + C)$

Simplify.

$\int_{1}^{2} e^{- 4 x} d x = - \frac{1}{4} (e^{- 8} - e^{- 4})$

Use the properties of exponents to further simplify the expression.

$\int_{1}^{2} e^{- 4 x} d x = \frac{e^{- 4} - e^{- 8}}{4}$

$\int_{1}^{2} e^{- 4 x} d x = \frac{e^{- 8} (e^{4} - 1)}{4}$

$\int_{1}^{2} e^{- 4 x} d x = \frac{e^{4} - 1}{e^{8}}$

Common Mistakes When Integrating Exponential Functions

We might get tired at a certain point after practicing for a while. This is where mistakes start to show up! Let's take a look at some common mistakes that we might make when integrating exponential functions.

We have seen a shortcut for integrating exponential functions when their argument is a multiple of x.

$\int e^{a x} d x = \frac{1}{a} e^{a x} + C$

This saves us plenty of time for sure! However, one common mistake is multiplying by the constant rather than dividing.

$\int e^{a x} d x \neq a e^{a x} + C$

This might happen to you if you just differentiated an exponential function, maybe you were doing Integration by Parts.

The following mistake concerns every antiderivative.

Another common mistake when integrating (not only exponential functions!) is forgetting to add the integration constant. That is, forgetting to add +C at the end of the antiderivative.

Always make sure to add +C at the end of an antiderivative!

$\int e^{x} d x = e^{x} + C$

Summary

Integrals of Exponential Functions - Key takeaways

The antiderivative of the exponential function is the exponential function itself. That is:∫exdx=ex+C
- If the argument of the exponential function is a multiple of x then: $\int e^{a x} d x = \frac{1}{a} e^{a x} + C$ where $a$ is any real number constant other than 0.
Two useful limits for evaluating improper integrals involving exponential functions are the following:
- $\lim_{x \to - \infty} e^{x} = 0$
- $\lim_{x \to \infty} e^{- x} = 0$
You can involve different Integration Techniques when finding the integrals of exponential functions.

Frequently Asked Questions about Integrals of Exponential Functions

The integral of the exponential function is an exponential function with the same base. If the exponential function has a base other than e then you need to divide by the natural logarithm of that base.

You can use methods like Integration by Substitution along with the fact that the antiderivative of an exponential function is another exponential function.

Since the half-life exponential decay function is an exponential function, its integral is another function of the same type.

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Integrals of Exponential Functions

Integrals of Exponential Functions

Definite Integrals of Exponential Functions

Integrals of Exponential Functions Examples

Common Mistakes When Integrating Exponential Functions

Summary

Integrals of Exponential Functions - Key takeaways

Frequently Asked Questions about Integrals of Exponential Functions

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Integrals of Exponential Functions

Integrals of Exponential Functions

Definite Integrals of Exponential Functions

Integrals of Exponential Functions Examples

Common Mistakes When Integrating Exponential Functions

Summary

Integrals of Exponential Functions - Key takeaways

Frequently Asked Questions about Integrals of Exponential Functions

What is the integral of an exponential function?

How to calculate integrals of exponential functions?

What is the integral of the half-life exponential decay function?

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