Graphs of indefinite integrals

- [Instructor] Find the general indefinite integral. And so we have the integral of two x dx. Which of the graphs shown below, which of the graphs below shows several members of the family? So if we're talking about, so if we're taking the integral of two x dx, we're talking about the antiderivative of two x. And what's that going to be? Well, it's going to be two x to the second power. 'Cause this was two x to the first power, so we increment the exponent to two. And then we divide by that newly incremented exponent, so this is going to be x squared. And you might have done that on your own, you said, okay, I know that the derivative of x squared is two x, so the antiderivative of two x is x squared. But we aren't quite done yet. Because remember, this isn't the only antiderivative of this. We could add any constant here. If we add some constant here and we take the derivative of it, we still get two x. Because the derivative of a constant with respect to x, it's not changing with respect to x, so its derivative is zero. So the antiderivatives, I guess you could say here, take this form, take the form of x squared plus C. Now what does that mean visually? So, let me draw, I can draw a neater version of that. So slightly better. So if that's my y-axis and this is my x-axis, we know what y equals x squared looks like. Y equals x squared looks like, I'll just draw the general shape. So y equals x squared looks like this. Now, what happens if I add a C? Let's say if I add, let's say y is equal to x squared plus two. And two is a valid C, so we could say, so let me write this down. This right over here is y is equal to x squared. But remember, and I guess you could say that, in this case, our C is zero. But what if our C was some positive value? So let's say it is y is equal to x squared plus, I don't know, y is equal to x squared plus five. Well, then we're going to have a y-intercept here at five, so essentially we're just gonna shift up the graph by our constant right over here, which is positive five. So we shift up by positive five, and we will get something that looks like this. We just shifted it up. Now, you might be saying, okay, well, that kind of looks like this choice right over here. But this choice also, also has some choices that start down here, that we're adding a constant. But you remember, this constant can be any constant. It could be a negative value. So in this case, C is five. In this case, C is zero. But C could also be negative five. So C could also be negative five. So if we wanted to do y is equal to x squared plus negative five, which is really x squared minus five, then the graph would look like this. It would shift x squared down, down by five. So this one is shifted up by five. This one is shifted down by five. So you would shift by the constant. If it's a positive constant, you're going up. If it's a negative constant, you are going down. So B is definitely the class of solutions to this indefinite integral. You take any, any of the functions that are represented by these graphs, you take their derivative, you're going to get two x. Or another way to think about it, the antiderivative of this or the integral, the indefinite integral of two x dx is gonna be x squared plus C, which would be represented by things that look like, so essentially things, essentially y equals x squared shifted up or down. So I could keep drawing over and over again.