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Antiderivatives

Suppose we have a function g(x) which we know is the derivative [with respect to x] of some other function f(x), but we don't know which - i.e. we know g(x) explicitly but we don't know [yet] what f(x) it is the derivative of. We may then ask the question, "What is the function f(x) whose derivative [with respect to x] is g(x)?" Another way of putting this would be to ask, "What is the antiderivative of g(x)?"^11.1 Another name for the antiderivative is the integral, which is in fact the "official" version, but I like the former better because the name suggests how we go about "solving" one.^11.2

For a handy example consider   $g(x) = k \, x$.  Then the antiderivative [integral] of g(x) with respect to x is   $f(x) = {1\over2} \, k \, x^2 \; + \; f_0$  [where  f₀  is some constant] because the derivative [with respect to x] of x² is 2x and the derivative of any constant is zero. Since any combination of constants is also a constant, it is equally valid to make the arbitrary constant term of the same form as the part which actually varies with x, viz.   $f(x) = {1\over2} \, k \, x^2 \; + \; {1\over2} \, k \, x_0^2$. Thus f₀ is the same thing as ${1\over2} \, k \, x_0^2$ and it is a matter of taste which you want to use.

Naturally we have a shorthand way of writing this. The differential equation

$$df = g(x) \; dx$$

can be turned into the integral equation

$$f(x) = \int_{x_0}^x g(x) \; dx$$ (11.2)

which reads, "f(x) is the integral of g(x) with respect to x from x₀ to x." We have used the rule that the integral of the differential of f [or any other quantity] is just the quantity itself,^11.3 in this case f:

$$\int df = f$$ (11.3)

Our example then reads

$$\int_{x_0}^x k \, x \, dx = k \int_{x_0}^x x \, dx = {1\over2} \, k \, x^2 \; - \; {1\over2} \, k \, x_0^2$$

where we have used the feature that any constant (like k) can be brought "outside the integral" - i.e. to the left of the integral sign   ${\displaystyle \int}$.

Now let's use these new tools to transform Newton's SECOND LAW into something more comfortable.