Graphs of indefinite integrals

Find the general indefinite integral.

So we have the integral of 2x 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, [Music] 2x dx, we're talking about the anti-derivative of 2x.

And what's that going to be?

Well, it's going to be 2x to the 2nd power because this was 2x to the 1st power.

So we increment the exponent to 2 and then we divide by that newly incremented exponent.

So this is going to be x².

And you might have done that on your own.

You said, "Okay, I know that the derivative of x² is 2x."

So the anti-derivative of 2x is x².

But we aren't quite done yet because remember this isn't the only anti-derivative of this.

We could add any constant here.

If we add some constant here and we take the derivative of it, we still get 2x 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 anti-derivatives, I guess you could say here, take this form; take the form of x² + C.

Now what does that mean visually?

So let me draw, I can draw a near version of that, so slightly better.

So if that's my y-axis and this is my x-axis, we know what y = x² looks like.

y = x² looks like this.

Now what happens if I add a constant?

If I add a C, let's say, if I add, let's say, y is equal to x² + 2, and 2 is a valid C, so we could say, so I'm going to write this down:

This right over here is y = x².

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 = x² + 5.

Well then we're going to have a y-intercept here at five.

So essentially, we're just going to 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 has some choices that start down here.

I thought 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 -5.

So C could also be -5.

So if we wanted to do y = x² + (-5), which is really x² - 5, then the graph would look like this; it would shift x² 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 the negative constant, you are going down.

So B is definitely the class of solutions to this indefinite integral.

You take any of the functions that are represented by these graphs, you take their derivative, you're going to get 2x.

Or another way to think about it, the anti-derivative of this, or the integral; the indefinite integral of 2x dx is going to be x² + C, which would be represented by things that look like, so essentially, things essentially y = x² shifted up or down.

So I could keep drawing over and over again.