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find the general indefinite integral and 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 2x DX we're talking about the antiderivative of 2x and what's that going to be well it's going to be 2x to the 2nd power because this was 2 X to the first power so we increment the exponent to 2 and then we divide by the 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 2x so the antiderivative of 2x 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 we take the derivative of it we still get 2 X because the derivative of a constant with respect to X it's not changing with respect to X so it's derivative is 0 so the anti derivatives 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's 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 if I add AC let's say if I add let's say y is equal to x squared plus 2 when 2 is a valid C so we could say so I'm going to 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 RC with some positive values 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 we're 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 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 0 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 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 to X or another way to think about it the antiderivative of this or the integral the indefinite integral of 2x DX is going to be x squared plus C which would be represented by things that look like so essentially things such a y equals x squared shifted up or down so I could keep drawing over and over again