A definite integral that defines a constant, and solving it by iteration

Last time

Before we start: last time, in Iteration, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

The constant hides in the limit

Here is a question type that chains three skills together. A definite integral from zero to a is given as a number. Integrate by parts, and you get an equation that a satisfies. But that equation has a on both sides, with an exponential, so you cannot solve it directly. Show the root lies in an interval, then iterate.

Plan of attack

Part one: integrate by parts, apply the limits, and rearrange to a given equation. The answer is printed, so every line counts. Part two: verify that a lies between two values by evaluating the equation's difference at both and spotting a sign change. Part three: iterate the formula until two successive values agree to the required accuracy.

Integration by parts

Let u equal x and v prime equal e to the minus 2x. Then u prime is 1 and v is minus a half e to the minus 2x. The integral of x e to the minus 2x is minus x e to the minus 2x over 2, plus a half of the integral of e to the minus 2x. Finish: minus a half x e to the minus 2x, minus a quarter e to the minus 2x.