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we are now going to talk about integration using initial conditions and particular solutions if you take a look at the graph here youll notice these are all the same shape what we have here are the general solutions of an antiderivative the antiderivative is X cubed minus X minus 2 and if you take a look at these graphs they all obviously are cubic graphs notice if you look closely though all of these curves although they have the same shape pass through different points on the y-axis thats because the constant of integration the C value would be different for each one of these curves the variable part is identical for all of them but what differs is the constant which of course we know affects the height of the curve in the coordinate plane youll notice one of these curves running right here through the point 0 negative 2 is in boldface that is the case in which we were given an initial condition specifically that the point 2 comma 4 had to lie on the original curve you can see it
