Integrals of Exponential Functions: Videos & Practice Problems

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To find the integral of an exponential function $b^x$, divide by the natural log of the base: $\frac{b^x}{\ln }$. For $e^x$, the integral is $e^x$ since $\ln e$ equals 1. Use the constant multiple rule to simplify integrals involving constants. Apply these rules to solve complex integrals, ensuring to add the constant of integration $c$.

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Integrals of General Exponential Functions

At this point in the course, we've learned a ton of different rules for finding the integrals of various functions. We've learned a lot of those rules based on the idea that taking an integral is really just the reverse process of taking a derivative. Now that we're faced with finding the integral of our general exponential function, b to the power of x, we're going to use that same idea here. So let's jump right into things. Recall that when finding the derivative of our general exponential function, again, that's b to the power of x, we multiplied bx by the natural log of b.

So when finding the derivative of something like 7x, this gives us 7x times the natural log of seven. Now, when faced with finding the integral of functions like this 7x, we just need to reverse this process. So when finding the integral of any general exponential function, we are now going to be dividing bx by the natural log of b instead of multiplying. So, in our rule box here, when we take the integral of bxdx, this is going to give us bxln b. Since this is an indefinite integral, we need to add the constant of integration c.

Now looking at this rule, we have certain restrictions on our base b that it must be greater than zero and not equal to one, but these are just restrictions based on the definition of a general exponential function and aren't typically something you're going to have to worry about. So let's apply this rule to our integral here when we integrate bxdx. Here, I can see that I have this base of seven. So with b equal to seven, applying my rule here, this is going to give me 7xln 7. Then, of course, plus that constant of integration, c.

Having reversed our rule for finding the derivative to now find the integral, we can use all the rules for integrals that we've already learned along with this new rule, so let's work through one additional example. Here we're asked to evaluate the indefinite integral of three times one half to the power of x plus eight to the power of x d x. Depending on how comfortable you are with integrals at this point, you may be able to do this all in one go.

But, I'm going to go ahead and break this down step by step just so you can see what I'm doing along the way. Since I have two different functions being added together here, I'm going to split this into two integrals being added together using my sum rule. This is going to be equal to the integral of three times one half to the power of x d x, and then plus the integral of eight to the x d x. Now I can do one more thing here to simplify this. I can use my constant multiple rule to pull that three out front of the integral, making this three times the integral of one half to the power of x d x.

Looking at this first term here, taking that constant, I'm going to multiply three times the integral of one half to the power of x d x. In applying this rule here, I'm just plugging in that one half everywhere that b is. This is going to be equal to 0.5xln 0.5.

Then adding that together with my other integral here, I can see that I have a base of eight. So plugging eight in where b is for my rule here, this is going to give me 8xln 8. Since we're working with indefinite integrals here, I need to add the constant of integration, c. Now, this is already an acceptable answer, but I can rewrite certain things here slightly based on exponential and log properties, and this may be something that's expected of you. So let's rewrite this using our exponential and log properties.

One half to the power of x can rewrite as one and two in my numerator and denominator to the power of x. This gives me three times one to the power of x over two to the power of x. Then when rewriting this natural log of one half, I could rewrite this as natural log of two to the power of negative one.

Keeping all my other terms the same here, eight to the power of x over the natural log of eight plus that constant of integration c. Simplifying this to its most simple form, one to the power of x is just always going to be one. So my numerator here is just three. Then I'm going to move that two to the power of x to the bottom. Then the natural log of two to the power of negative one, I can pull that negative one out front using my log property here to make this whole term negative.

This becomes negative three over two to the power of x times the natural log of two. Then adding on my other term here, keeping it the same, eight to the power of x over the natural log of eight plus that constant of integration c. Now we've fully simplified this using our exponential and log properties, and this is our final answer. We're going to continue getting practice with this new rule coming up in the next couple of videos. I'll see you there.

Evaluate the indefinite integral. 
$\int_{}^{}-\left(6\right)^{x}dx$

• A. $-x6^{x-1}+C$
• B. $-\frac{1}{x+1}6^{x+1}+C$
• C. $-\frac{\left(6\right)^{x}}{\ln6}+C$
• D. $-\frac{\left(6\right)^{x+1}}{\ln x}+C$

Answer: (C)

Evaluate the indefinite integral.
$\int_{}^{}3x^4-\left(5\right)^{x}dx$

• A. $\frac35x^5-\frac{5^{x+1}}{\ln5}+C$
• B. $\frac35x^5-\frac{5^{x}}{\ln5}+C$
• C. $12x^5-\frac{5^{x}}{\ln5}+C$
• D. $12x^5-\frac{5^{x+1}}{\ln5}+C$

Answer: (B)

Evaluate the indefinite integral.
$\int_{}^{}2\left(\frac13\right)^{x}dx$

• A. $-\frac{2}{3^{x}\ln3}+C$
• B. $-\frac{2x^{x+1}}{4}+C$
• C. $-\frac{2x}{\ln3}+C$
• D. $-\frac{2x^{x+1}}{3^{x}\ln4}+C$

Answer: (A)

Integrals of Natural Exponential Functions (e^x)

In the previous video, we learned how to find the integral of our general exponential function b raised to the power of x. Now, a common exponential function that we'll encounter is e raised to the power of x. So how do we then find this integral? Well, luckily, since e to the x is really just a special case of b to the x where our base is just e, we can use the integral rule that we just learned for our general exponential functions to now find the integral of e to the x. So that's exactly what we're going to do here.

Let's go ahead and jump right into things. Now looking at this integral exdx, we want to look to our rule for b to the x. And since our base here is e, we just want to replace all of these instances of b with e. So taking this integral is then going to give me exln(e). Now looking at this, I know that the natural log of e is really just one.

So ex1 just gives me ex. So I'm left here with ex+c, since we're dealing with an indefinite integral. So integrating e to the x just gives me e to the x. I'm left with the same function that I started with. Now this makes sense because we saw the same thing when taking the derivative of e to the x.

So what does this integral rule look like in action? Well, here we're faced with the integral of five times e to the x dx. Now five is just a constant. So I know using the constant multiple rule, I can just pull it out front, making this five times the integral of e to the x dx. Now based on our rule here, we will know that the integral of exdx is just ex.

So this ultimately gives me five times ex and then, of course, plus that constant of integration c since this is an indefinite integral. So now that we have this rule, we can use this along with all of our other integral rules that we've learned so far to find the integral of some more complicated functions that may now involve e. So looking at this integral here, we want to integrate three x raised to the power of four minus pi times e raised to the x plus two dx. Now, depending on how comfortable you are with integrals at this point, you can do this all in one go. But I'm going to go ahead and break this integral down just so you can see what I'm doing step by step.

The first thing I'm going to do here is break this into three separate integrals using my sum and difference rules. So this becomes the integral of three x raised to the power of four dx minus the integral of pi times e to the x dx, and then plus the integral of two dx. So now we can focus on each of these integrals separately. So focusing on this first one here, the integral of three times x raised to the power of four dx. Three is just a constant.

So using my constant multiple rule, I know that I can pull this three out front. And just look at the integral of x raised to the power of four dx, which using the general power rule is going to be one fifth x raised to the power of five. So with that three, this is going to be three fifths x raised to the power of five. Then looking at our next integral here, pi times e to the x dx. Now remember that pi is a constant.

It is not a variable. So I can use my constant multiple rule here as well. Pulling that pi out front, focusing just on integrating e raised to the power of x dx. And based on the rule that we just learned, we know that that just leaves us with ex. So this is just pi times ex, the same function that we started with.

Then our final integral here, two dx two is just a constant. So this is going to give me two x, and then, of course, plus that constant of integration c on the end of my integral here. So this is my final answer, having applied multiple different rules for integration. Now we're going to continue getting practice with taking integrals and all of the rules that we've learned so far coming up next. I'll see you there.

Evaluate the indefinite integral.
$\int_{}^{}\left(\sqrt{x}-e^{x}\right)dx$

• A. $\frac23x^{\frac23}-e^{x}+C$
• B. $\frac23x^{\frac32}-xe^{x}+C$
• C. $\frac23x^{\frac32}-e^{x}+C$
• D. $\frac23x^{\frac23}-\frac{1}{x}e^{x}+C$

Answer: (C)

Evaluate the indefinite integral.
$\int_{}^{}\left(4e^{x}+\frac{1}{x^3}\right)dx$

• A. $4e^{x}-\frac{1}{4x^4}+C$
• B. $4xe^{x}-\frac{1}{2x^2}+C$
• C. $4xe^{x}-\frac{1}{4x^4}+C$
• D. $4e^{x}-\frac{1}{2x^2}+C$

Answer: (D)

Here’s what students ask on this topic:

What is the integral of an exponential function b^x?

The integral of an exponential function b^x is given by the formula: $\frac{b^x}{\ln \left(b\right)}$ plus a constant of integration, c. This formula is derived from reversing the process of differentiation for exponential functions. It is important to note that the base b must be greater than zero and not equal to one. This rule allows us to integrate functions like 7^x or 2^x by simply applying the formula and adding the constant of integration.

How do you find the integral of e^x?

The integral of e^x is straightforward because e^x is a special case of the general exponential function b^x where the base b is e. The integral of e^x is simply e^x plus a constant of integration, c. This is because the natural logarithm of e, ln(e), is equal to 1, simplifying the formula for the integral of b^x. Therefore, integrating e^x results in the same function, e^x, plus the constant c.

What is the rule for integrating exponential functions with different bases?

To integrate exponential functions with different bases, use the rule: $\frac{b^x}{\ln \left(b\right)}$ plus a constant of integration, c. This rule applies to any exponential function b^x where the base b is greater than zero and not equal to one. The process involves dividing the exponential function by the natural logarithm of its base. This rule is essential for integrating functions like 3^x or 5^x, allowing you to find the antiderivative by applying the formula and adding the constant c.

How do you apply the constant multiple rule in integration?

The constant multiple rule in integration states that if you have a constant multiplied by a function, you can factor out the constant and integrate the function separately. For example, if you have an integral of the form $kf\left(x\right)dx$, where k is a constant, you can rewrite it as $kf\left(x\right)dx$. This simplifies the integration process, allowing you to focus on finding the integral of the function f(x) and then multiplying the result by the constant k. This rule is particularly useful when dealing with integrals involving exponential functions or polynomials.

What are the restrictions on the base b for integrating exponential functions?

When integrating exponential functions of the form b^x, the base b must satisfy certain restrictions: it must be greater than zero and not equal to one. These restrictions are based on the definition of a general exponential function. A base of b = 1 would result in a constant function, which does not exhibit exponential growth or decay. Similarly, a base less than or equal to zero would not fit the standard definition of an exponential function. These conditions ensure that the function behaves as expected and that the integration rule $\frac{b^x}{\ln \left(b\right)}$ plus c is valid.