6/16/2016 > Math

The Bicycle Integral

A few posts ago I wrote about The Bicycle Problem. Specifically, I found a differential equation that modeled the path for a bicycle being pulled along a line. This differential equation involved solving a certain integral. After a few weak tries at solving it, I gave up and asked Wolfram-Alpha:
"I spent awhile trying to crack this one [integral], but considering what Wolfram-Alpha spit out, it would have take a lot longer than 'awhile'"
That sounds like I gave up. Which, in a sense, is true.

However, I don't want to sound like I rely on Wolfram-Alpha to do my math for me, so I decided to conquer this unassuming indefinate integral myself: \[\int -\frac{\sqrt{a^2-x^2}}{x}\,\,dx\]
Warning! Heavy use of calculus ahead!
Here's the graph of \(-\frac{\sqrt{a^2-x^2}}{x}\), for reference:
At first glance, a nice U-substitution looked promising, but the feeling only lasted for about 5 seconds: \[u=a^2-x^2 \qquad -\int \frac{\sqrt{u}}{-2x \cdot x}\,\,du\] Then I tried integrating by parts. If you don't recall, integrating by parts is about finding something in the function that you know the integral of, then using the product rule backward to work from there. For example, I know that the integral of \(\frac{1}{x}\) is \(\ln{x}\). Then I could set up an expression like this: \[-\ln{x}\cdot \sqrt{a^2-x^2}\] If I took the derivative of this using the product rule, I would get \[-\frac{d}{dx}(\ln{x})\cdot \sqrt{a^2-x^2} + -\frac{d}{dx}\left(\sqrt{a^2-x^2}\right)\cdot \ln{x}\] \[-\frac{\sqrt{a^2-x^2}}{x} - \frac{-2x\cdot \ln{x}}{2\sqrt{a^2-x^2}}\] And look at that! We get our original function. Unfortunately, we also get this other expression, \(\frac{2x\cdot \ln{x}}{2\sqrt{a^2-x^2}}\). To fix this, we need a new function with a derivative equal to \(-\frac{2x\cdot \ln{x}}{2\sqrt{a^2-x^2}}\) that, when added to the original function, would cancel out this term. And now we have a new integral to solve: \[\int -\frac{x\ln{x}}{\sqrt{a^2-x^2}}\,\,dx\] So altogether, we have \[\int -\frac{\sqrt{a^2-x^2}}{x}\,\,dx = -\ln{x}\sqrt{a^2-x^2} - \int \frac{x\ln{x}}{\sqrt{a^2-x^2}}\,\,dx\] But... That doesn't help us very much. This integral actually looks worse than the old one! In case you are looking at this thinking that you know the integral to \(\frac{x}{\sqrt{a^2-x^2}}\), here's what happens when you integrate by parts again: \[\int -\frac{\sqrt{a^2-x^2}}{x}\,\,dx = -\ln{x}\sqrt{a^2-x^2} - \int \frac{x\ln{x}}{\sqrt{a^2-x^2}}\,\,dx\] \[\int -\frac{\sqrt{a^2-x^2}}{x}\,\,dx = -\ln{x}\sqrt{a^2-x^2} - \left(-\sqrt{a^2-x^2}\ln{x} - \int -\frac{\sqrt{a^2-x^2}}{x}\,\,dx\right)\] \[\int -\frac{\sqrt{a^2-x^2}}{x}\,\,dx = \int -\frac{\sqrt{a^2-x^2}}{x}\,\,dx\] \[0=0\] Did you see that coming? If you think about it, you basically just undid what your first integration-by-parts accomplished, so you are back where you started.
So now what? At this point in the bicycle problem, after trying multiple other integration-by-parts attemps, I gave up and asked Wolfram-Alpha what the answer was. Not this time!

Remember when your pre-calculus teacher told you to always rationalize the denominator?

The lesson would look something like this: \[\frac{\sqrt{x}}{\sqrt{y}} = \frac{\sqrt{x}\sqrt{y}}{\sqrt{y}\sqrt{y}} = \frac{\sqrt{xy}}{y}\] Well, what if we un-rationalized the expression? \[-\frac{\sqrt{a^2-x^2}}{x} = -\frac{\sqrt{a^2-x^2}\sqrt{a^2-x^2}}{x\sqrt{a^2-x^2}} = -\frac{a^2-x^2}{x\sqrt{a^2-x^2}}\] Distributing the negative and dividing up the fraction: \[-\frac{a^2-x^2}{x\sqrt{a^2-x^2}}\] \[\frac{x^2}{x\sqrt{a^2-x^2}} - \frac{a^2}{x\sqrt{a^2-x^2}}\] \[\frac{x}{\sqrt{a^2-x^2}} - \frac{a^2}{x\sqrt{a^2-x^2}}\] Now there's something we can work with! The whole first term is now the U-Substitution I was looking for at the beginning (which I will leave as an excercise to the reader). \[\int \frac{x}{\sqrt{a^2-x^2}} - \frac{a^2}{x\sqrt{a^2-x^2}} \,\, dx = -\sqrt{a^2-x^2} + C - a^2\int \frac{1}{x\sqrt{a^2-x^2}}\,\,dx\] And so I stared at this new integral for a long time. And then...! Remember your Calculus I class, where you learned the derivatives for inverse trigonometric functions? Well, if you don't, there's one in particular that I thought of: \[\frac{d}{dx}\,\,\sec^{-1}{x} = \frac{1}{|x|\sqrt{x^2-1}}\] There is a certain similarity between this and my integral. In fact, with U-substitution: \[u=\frac{x}{a} \qquad x=au \qquad \int \frac{1}{au\sqrt{a^2-a^2u^2}}a\,\,du\] \[\int \frac{1}{au\sqrt{1-u^2}}\,\,du\] \[\frac{1}{a} \int \frac{1}{u\sqrt{1-u^2}}\,\,du \] There's just one more change that needs to be done: \[\frac{1}{a} \int \frac{1}{u\sqrt{-1\cdot(u^2-1)}}\,\,du \] \[\frac{1}{a} \int \frac{1}{iu\sqrt{u^2-1}}\,\,du \] \[\frac{1}{ia} \int \frac{1}{u\sqrt{u^2-1}}\,\,du \] And voila! Now we can set up a piecewise function to deal with the absolute value in the \(\sec^{-1}(x)\) derivative. \[\frac{1}{ia} \int \frac{1}{u\sqrt{u^2-1}}\,\,du = \begin{cases} \frac{1}{ia}\sec^{-1}{u} \quad \text{if }u >= 0 \\ -\frac{1}{ia}\sec^{-1}{u} \quad \text{if }u < 0 \end{cases}\] For simplicity sake, I'm going to stick with only \(u>=0\) for now.

Technically, we're done. The final product is \[\int -\frac{\sqrt{a^2-x^2}}{x}\,\,dx = -\sqrt{a^2-x^2} - a^2\left(\frac{1}{ia}\sec^{-1}{\left(\frac{x}{a}\right)}\right)+ C\] \[\int -\frac{\sqrt{a^2-x^2}}{x}\,\,dx = -\sqrt{a^2-x^2} - \frac{a}{i}\sec^{-1}{\left(\frac{x}{a}\right)}+ C\] However, we can't exactly type this expression into our favorite graphing program. We're going to have to get rid of that \(i\). This is where things get a little tricky.

Transforming \(\sec^{-1}{\left(x\right)}\) into basic functions (involving \(i\))

Let's start with Euler's Formula: \[e^{ix}=\cos{x}+i\sin{x}\] Rewriting in terms of cos: \[e^{ix}=\cos{x}+i\sqrt{1-\cos^2{x}}\] Rewriting in terms of sec: \[e^{ix}=\frac{1}{\sec{x}}+i\sqrt{1-\frac{1}{\sec^2{x}}}\] Substituting \(\sec^{-1}{\left(x\right)}\) for x: \[e^{i\sec^{-1}{\left(x\right)}} = \frac{1}{x} + i\sqrt{1-\frac{1}{x^2}}\] \[\sec^{-1}{\left(x\right)} = \frac{1}{i}\ln{\left( \frac{1}{x} + i\sqrt{1-\frac{1}{x^2}} \right)}\] \[\sec^{-1}{\left(x\right)} = \frac{1}{i}\ln{\left( \frac{1}{x} + \sqrt{\frac{1}{x^2} - 1} \right)}\] \[\sec^{-1}{\left(x\right)} = \frac{1}{i}\ln{\left( \frac{1}{x} + \frac{1}{x}\sqrt{1-x^2} \right)}\] \[\sec^{-1}{\left(x\right)} = -i\left(\ln{\left( 1 + \sqrt{1-x^2} \right)} + \ln{\left(\frac{1}{x}\right)}\right)\]
And, substituting that into our previous equation: \[\int -\frac{\sqrt{a^2-x^2}}{x}\,\,dx = -\sqrt{a^2-x^2} - \frac{a}{i}\cdot-i\left(\ln{\left( 1 + \sqrt{1-\left(\frac{x}{a}\right)^2} \right)} + \ln{\left(\frac{1}{\frac{x}{a}}\right)}\right)+ C\] \[\int -\frac{\sqrt{a^2-x^2}}{x}\,\,dx = -\sqrt{a^2-x^2} +a\left(\ln{\left( 1 + \sqrt{1-\left(\frac{x}{a}\right)^2} \right)} + \ln{\left(\frac{a}{x}\right)}\right)+ C\] \[\int -\frac{\sqrt{a^2-x^2}}{x}\,\,dx = -\sqrt{a^2-x^2} +a\ln{\left( 1 + \sqrt{1-\left(\frac{x}{a}\right)^2} \right)} + a\ln{a} - a\ln{x}+ C\] Because \(a\ln{a}\) is a constant, we'll just lump it in with \(C\): \[\int -\frac{\sqrt{a^2-x^2}}{x}\,\,dx = -\sqrt{a^2-x^2} +a\ln{\left( 1 + \sqrt{1-\left(\frac{x}{a}\right)^2} \right)} - a\ln{x}+ C\] And there we go! It's finished! Now I feel better. I can do math.
Note: I neglected some domain issues throughout this whole process. While in that sense it isn't entirely rigorous, it all works out given that \(x>0\), which was true in the original bicycle problem.