IIRC, you take the derivative of the ‘inner’ function (x/2) and multiply it by the derivative of the ‘outer’ function (e[sup]x/2[/sup]). So the derivative of the ‘inner’ function is (1/2) * x[sup]0[/sup] = (1/2) * 1 = 1/2 and the derivative of the ‘outer’ function is e[sup]x/2[/sup], and the derivative of e[sup]x/2[/sup] is (1/2)*e[sup]x/2[/sup], right?

You’re right that the derivative of e[sup]x/2[/sup] is (1/2)e[sup]x/2[/sup]. But that means that the indefinite integral, or antiderivative, of (1/2)e[sup]x/2[/sup] is e[sup]x/2[/sup] + C, and then the integral of e[sup]x/2[/sup] is 1/(1/2)e[sup]x/2[/sup] + C, or 2e[sup]x/2[/sup] + C.

OK, how do you get from ∫e[sup]x/2[/sup] to 2e[sup]x/2[/sup] + C? (I know where the C comes from, since I didn’t include limits.)

Wait, let me think… If the coefficient of e is one, then to get the derivative 1/2 the coefficient of e needs to be 2. But how do I get there logically?

Also: Why do I get these things in my head and spend time on them?

The rules for derivatives are all well-established and algorithmic, such that any standard function or combination of standard functions has a derivative that is a combination of standard functions, and that combination can be found straightforwardly. The same is not, however, true of integrals. So when one is presented with a pair of functions and the claim that the one is the integral of the other, the simple way to check this claim is to perform the derivative, not to perform the integral.