I hope you enjoyed the previous activities about applications of integrals. I designed these activities to help you understand integration without having to do any tedious calculations. [br][br]You probably noticed however that the antiderivatives always seemed to appear "out of nowhere." For instance, in the [url=https://www.geogebra.org/m/x39ys4d7#material/yqenataa]previous activity[/url], when we integrated [code]v(x)=-9.8x+v_0[/code], there was almost no fanfare regarding the appearance of the antiderivative [code]s(x)=-4.9x^2+v_0*x+s_0[/code]. Similarly, when we studied the [url=https://www.geogebra.org/m/x39ys4d7#material/tx9hrc7k]intuitive proof of the FTC[/url] by taking a close look at the integral of the traffic on Route 15, the antiderivative just appeared. The reason for this was that I didn't want you to take your eye off the prize: building conceptual knowledge about the integral. [br][br]It's time however to talk about a process to find antiderivatives. This process amounts to undoing the [url=https://www.geogebra.org/m/x39ys4d7#material/p8jdmayj]Monkey Rules[/url] from earlier in the textbook. [br][br]The good news is that conceptually, this is a good bit easier than what we've done so far in this chapter on integrals.[br][br]The bad news is that this is a little harder than using the Monkey Rules to calculate derivatives. In some sense the Monkey Rules, particularly the [url=https://www.geogebra.org/m/x39ys4d7#material/jywee6u7]Quotient Rule[/url] and the [url=https://www.geogebra.org/m/x39ys4d7#material/kep6gkad]Chain Rule[/url], "blow functions up" when they systematically calculate derivatives. In order to go backwards, and undo the Monkey Rules to find antiderivatives, you need to think a bit like a forensic analyst who studies the site of an explosion to see what sort of bomb was used. We'll discuss this analogy more later when we practice finding antiderivatives.[br][br]Because of the new level of difficulty of attempting to do the Monkey Rules backwards, I call the rules for finding antiderivatives [b]Lucifer's Rules[/b]. [br][br]In general, your ability to calculate antiderivatives doesn't really impact your conceptual understanding of calculus. However, if you plan to continue your study of calculus, you do need to have at least a passing understanding of the fundamentals of finding antiderivatives. In future seasons we'll need a little bit of skill finding antiderivatives.[br][br]One last word before we get started: As we temporarily move away from studying integrals as an applied topic, we need some notation for antiderivative. After all, when an integral is calculated, you end up with a [i]number[/i], but an antiderivative is a function. The standard notation for "the antiderivative of f(x)" is:[br][br][math]\int f\left(x\right)dx[/math][br][br]The only difference between this notation for "the antiderivative of [code]f(x)[/code]" and the notation for "the [url=https://www.geogebra.org/m/x39ys4d7#material/ufsyvbbx]integral of [code]f(x)[/code] from [code]x=a[/code] to [code]x=b[/code][/url][code][/code]" is the absence of the bounds. I apologize that it isn't more different, but this is standard, and I think that if I tried to invent my own notation, it would confuse you more than it would help you.