Question Video: Finding the Integration of an Exponential Function with an Integer Base Mathematics • Higher Education

Determine ∫ 2^(9𝑥) d𝑥.

01:22

Video Transcript

Determine the integral of two to the power of nine 𝑥 with respect to 𝑥.

Let’s begin by quoting what we do know about the integral of 𝑎 to the power of 𝑥. It’s 𝑎 to the power of 𝑥 divided by the natural log of 𝑎. Our integrand is slightly different though; it’s a constant to the power of another constant times 𝑥. So, we’re going to use the process of introducing something new, a new letter. We let 𝑢 be equal to nine 𝑥, and of course this is known as integration by substitution. We obtain the derivative of d𝑢 with respect to 𝑥 to be equal to nine. Now, remember, d𝑢 by d𝑥 is absolutely not a fraction, but we can treat it a little like one for the purposes of integration by substitution. And we see that we can say that a ninth d𝑢 equals d𝑥.

We replace 𝑢 with nine 𝑥 and d𝑥 with a ninth d𝑢. And then, we take out this constant factor of a ninth. And we see that our integral is now a ninth of the integral of two to the power of 𝑢 with respect to 𝑢. Well, the integral of two to the power of 𝑢 is two to the power of 𝑢 over the natural log of two. And then, of course, we can use the definition of our substitution and replace 𝑢 with nine 𝑥. And we’ve found the integral of two to the power of nine 𝑥 with respect to 𝑥. It’s two to the power of nine 𝑥 over nine times the natural log of two plus this constant of integration 𝐶, which I’ve made a capital 𝐶 to show that it’s different from the value we had before.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.