A hard integral computed easily

Equation 1.

In my previous article on doing integration with the help of inverse functions https://medium.com/cantors-paradise/integration-using-inverse-functions-12094a9bc508 I showed a method for integrating seemingly complicated functions over a finite domain, by simply computing the area of the inverse function and matching it to the area we actually want to compute. In this article, I will apply that same method for solving another interesting integral, the one shown in Figure 1.

First, we must think about the range of the function y = f(x) in the domain x in [0, 1] and what the function looks like. Plugging in a few values of x, we quickly realize that the function is monotonically decreasing, and its range is y = [0, 1] as well. In that sense, we can apply the following formula for computing the definite integral using the inverse function:

This formula is similar to what I presented in my last article, except that one assumed the function was monotonically increasing in the domain [a, b] rather than decreasing. Nevertheless, the procedure is pretty much the same as before. The inverse function in this case is

and this can be expanded into

which is a simple polynomial which can be integrated term by term. Then, we apply Equation 2 to find the desired value:

a value which you can verify yourself numerically or using WolframAlpha.

One thing I would like to point out before concluding this rather short article is this: sometimes difficult problems have easy solutions, which seem obvious when pointed out by someone else, but which are not so obvious when we are first confronted by them. I guess this is an inevitable aspect of human nature. The only sure way to become better at math, and problem solving in general, is to actually spend a lot of time reflecting about these kinds of results, thinking about what you can do by twisting and turning things around, rotating graphs, and just trying to imagine what each mathematical concept means in the grander scheme of things. Textbooks and math courses are good but they are limited, they only teach you standard content which has remained largely unchanged for the past 100 years, and without doing explorations of your own, you will not be able to gain full mastery of math.

As an exercise to the reader, I recommend looking again at Equation 2, and think to yourself: “is this true? under what conditions is it true? what does this look like when I plot it?”. Things will be much clearer when you get a geometric intuition for this.