The Two Main Antiderivative Rules
Special Case Antiderivatives (The ones you have to memorize)
U-Substitution
Initial Value Problems: Finding the +C
Fundamental Theorem of Calculus Part 1

Indefinite Integrals

An Indefinite Integral is an antiderivative in the truest sense of the word. You start with a given differential equation = derivative = f ( x ) = dy dx , and use your antiderivative rules (just like you had derivative rules ) to undo the derivative and get back the general solution = original equation = f ( x ) .

f ( x ) dx = f ( x ) + C

You will notice that the results in Indefinite Integral problems will always include a +C . What this means is that technically you are not getting one solution, you are getting what is often called a “family” of solutions. Essentially the “family” of equations, f ( x ) + C , all are the same function (the same graph) the only difference between each of them is a vertical shift, the +C . Since the only difference between all the members of a “family” is a constant, +C , the entire family has the same instantaneous rate of change, f ( x ) , at every x-value because the derivative of any constant, +C , is always zero. This means the constant, +C , could have been any value . As you do not know about the value constant when you are applying the antiderivative rules , you track that it could have been there, by always putting a +C at the end of the result.

For example, if you started with the derivative f ( x ) = 3 x 2 + 10 x 7 , the antiderivative result would look like 3 x 2 + 10 x 7 dx = x 3 + 5 x 2 7 x + C . Putting this into the context of a graph you can see how the idea of a family of solutions works. Changing the value of the +C does not change the shape of the graph, so it does not change its instantaneous rate of change. It merely shifts the same graph up or down.

Keep this “family” of solutions image in mind when you are doing slope fields later on. Slope fields are another method for visualizing the “family” of solutions.

You will also see application problems, referred to as initial value problems , where you are given an actual point , (x,y), that is from the original equation, f ( x ) , and you can use that point to find out an equations specific solution, its specific +C .

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