So if we want to find the demand function based on each of these marginal revenue functions, So we might need to write a couple of things down before we start this problem just to kind of guide us and what we're doing. So first, remember that marginal is a very small change of something. So what this is saying is that this marginal function is really just the derivative. You're too of our revenue function. So if we were to integrate our revenue function, we get back. I'm sorry. If we integrate our marginal revenue, we should get the revenue function back. But we don't want to find the actual revenue function. We want to find the demand. So you might recall that revenue is equal to demand times the number of items sold. So our is revenue, these demand. And actually, maybe I'll write this with the actual variables. So our X is equal to D of X, and then this will be times X. So But this tells us, or this implies that our demand function is the revenue function divided by next. And in this case, X is the number of items were selling. So what we first want to do is go ahead and integrate this function. So you go have him scoop this over a little bit. Mm. There's some a little bit. Mm, obviously. Oh, go ahead and put metrical sign here again. And so I want to first integrate this so I can solve for the revenue function. So integrating this with respect to X So doing that gives our revenue function are of X. And now I can go ahead and integrate the right hand side. So remember, if it's not written and you just have a constant, it's implied that you have a constant to the zero power being multiplied. So we'll go ahead and use the uh huh, some and scalar property of integration. And doing that will give so 50 next zero. So just first rewriting the expression next to the two thirds and then giving us a little space so we can write in our integration symbol so integral of DX here an integral DX there. And now both of these are power rules. So this will be 50 x to the zero plus one over the new power which will be one plus some constant of integration. See one that I'll call it and then minus bye X to the two thirds plus one. So two thirds plus one would be five thirds. And then I add another integration. Constant. Call it CT. So let me go ahead and speak to stuff a little bit. Just get myself a little bit more room. Yeah. Now, since we have to integration Constance here, we can go ahead and add them together because having two constants give another constant. So I'll go ahead and call this constant C and AL I can clean up the other part. So the first part will become 50 X and then five, divided by five thirds will be three. It would be three x to the five thirds power plus c, and now this year will be our revenue function. Right. So remember, we're still looking for demand function, So I want to end up dividing this by X eventually. But before we do that, let's solve for what our constant of integration C is so we can find out what our constant of integration is. If we use the second line in this problem where it says call that if no items are sold, the revenue zero. So this is really tired. Asses R of zero is equal to zero. So let's go ahead and plug this and or combined this fact with our revenue equation and doing that will give So zero is equal to all. Go ahead and write, child the expression first and the parentheses for my variable. Yeah, this class c and then I want to plug in zero for each of those. So I end up with zero is equal to so 50 times zero is zero and taking the five thirds power of 00 multiplied by 30 So this is also zero. So we just end up with C is equal to zero. Okay, Now, we can use this fact here to plug into our revenue equation and doing that, and they don't go ahead and actually move all this up here now. So doing that tells me that are of X is equal to 50 x minus three x to the five thirds power. Now to get the demand function, I want to divide this whole thing by X so d o x is equal to our of X over X and then dividing X into 50 x will give me 50 and then dividing X into three x five thirds will give so X to the two thirds power, and this here will be our demand equation.