Evaluate the following integrals.
∫ e³ˣ/(eˣ - 1) dx

Question

Evaluate the following integrals.

∫ e³ˣ/(eˣ - 1) dx

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to find the indefinite integral, and we're given the integral of E 4 X divided by the quantity of E X minus 1 and quantity DX. So, to evaluate this integral, we'll first use U substitution. So we'll set U equal to E X. And that means that DU is going to be equal to E X D X. So, our integral of E to the 4 X. Divided by the quantity of E to the X minus 1 in quantity DX is going to be equal to the integral of UQ divided by the quantity of U minus 1 and quantity DU. Where we have taken 1 power of ETDX in the numerator and combined it with DX to get our DU, and that leaves 3 powers of ETDX left over, and that's why we have you cubed in the numerator. Now, even in this form after doing the use substitution, our integral here cannot be integrated directly. So what we're going to do is use long division to remove our fraction. So we're going to divide U minus 1 into. You cued. And so, first step is to see how many times you will go into U cubed, and that will go U2 times. We'll have you squared multiplied by the quantity of U minus 1. This will give us U cubed minus U2. And we'll subtract that from you cubed, and this will give us U squared left over. And so now we'll have to divide U into U squad and that will go U times. So we'll have plus U in our quotient. So we multiply you by the quantity of U minus 1, which will give us U quad minusu. And we'll subtract that from you squared, and this will give us u. And so you will go into you one time, so we'll have plus 1 in our quotient, and so we'll have 1 multiplied by the quantity of 1, sorry, of U minus 1. And that will give us U minus 1, and we'll subtract that from you, and we'll have 1 as our remainder. So now we can rewrite our integral, and so we'll have the integral. Of U cubed, divided by the quantity of U minus 1 in quantity DU, that's going to be equal to the integral of the quantity. Of U squared plus U + 1. Plus 1 divided by the quantity of U minus 1 in quantity, and all of that is multiplied by DU. So now we just need to integrate each of these terms individually. Though our integral here is going to be equal to. For the first term, we're integrating U2d with respect to you, and that will give me you cubed divided by 3. Then we'll have plus for the next term we're integrating you. I want to recall that when I integrate you, I get U squared, divided by 2. Then we'll have plus. The next term we're integrating 1 with respect to you, which will give us you. And then we'll have plus, and for our last term here, we're actually gonna have to use substitution again. So we're gonna set W equal to U minus 1, so that means that DW is equal to DU. And so the integral of 1 divided by the quantity of 1 minus 1 in quantity DU is going to be equal to the integral of 1 divided by WDW, which is actually equal to the natural log of the absolute value of W plus C. So, for our last term, we're going to have the natural log of the absolute value, and instead of W we'll have you minus 1. And then we also have to have our integration constant from all four integrations, so we'll just combine those into a single constant of plus C. So now all we have to do is to substitute back in. Or um. ETDX for you, and so we'll see that we have that the integral of E to the 4 X divided by the quantity of E to the X minus 1 in quantity DX is going to be equal to. 1 divided by 3, multiplied by E to the 3 X. Plus, 1 divided by 2 multiplied by E to the 2 X. Plus e to the X. Plus the natural log of the absolute value of the quantity of E to the X minus 1 in quantity plus C. So this here is the answer that we are looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Evaluate the following integrals.
∫ e³ˣ/(eˣ - 1) dx

Key Concepts

Integration

Exponential Functions

Improper Integrals

Watch next

Evaluate the following integrals.∫ e³ˣ/(eˣ - 1) dx

Key Concepts

Integration

Exponential Functions

Improper Integrals

Watch next

Evaluate the following integrals.
∫ e³ˣ/(eˣ - 1) dx