The premise is the following:

We assume that we have a function f(x) which we can integrate over the interval a to b.

Calculate the following integral: $$\int_{2}^{4}\frac{\sqrt{x}}{\sqrt{6-x} + \sqrt{x}} dx$$ Doesn’t look trivial now, does it? But miraculously, it is! This whole integral equates to: $$\frac{b-a}{2} = 1$$.

Let’s take a look at another integral for fun. $$\int_{0}^{\frac{\pi}{2}}\frac{\sqrt{sin x}}{\sqrt{sin x} + \sqrt{cos x}}dx$$

Can you see an emerging pattern? This is yet again equal to: $$\frac{b-a}{2} = \frac{\frac{\pi}{2}}{2} = \frac{\pi}{4}$$.

We can observe that for any integral of the form $$\int_{a}^{b}\frac{f(x)}{f(a+b-x) + f(x)}dx$$ the following holds: $$\int_{a}^{b}\frac{f(x)}{f(a+b-x) + f(x)}dx = \frac{b-a}{2}$$.

Now if I were to just throw this out there I would probably get lynched by Ueli Maurer himself. Thus the proof goes as following: We know that: $$\int_{a}^{b}\frac{f(x)}{f(a+b-x) + f(x)}dx = I$$ This will be our first equation. Now we set \(u = a + b - x\) and \(\frac{du}{dx} = - 1 \implies du = -dx\) by putting that into our integral we receive: $$\int_{b}^{a}\frac{f(a+b-u)}{f(u) + f(a+b-u)} (-du) = I\newline \implies \int_{a}^{b}\frac{f(a+b-u)}{f(u) + f(a+b-u)} du = I \newline \implies \int_{a}^{b}\frac{f(a+b-x)}{f(x) + f(a+b-x)} dx = I $$ Since u is only used as a placeholder variable we may rename it as x. By adding these two integrals (which both equal I) together, we are left with: $$ \int_{a}^{b}\frac{f(a+b-x) + f(x)}{f(x) + f(a+b-x)} dx = 2I\newline \implies \int_{a}^{b} dx = 2I \newline \implies b-a = 2I \newline \implies I = \frac{b-a}{2} \square $$

Now you can solve some integrals very quickly and without the need of much calculation!

This post was largely inspired and based on this “MindYourDecisions” video. Check it out for another example for an integral which you can solve with this method.

]]>Algorithms and Probabilities, or AnW for short, is all about probabilistic algorithms. This ranged from colouring to matchings over to probabilities (expected value, variance) to flows and convex hulls.

Analysis I is your basic Calculus course. From supremum and infimum up to partial integration this represents the main, purely mathematical part of this semester’s courses.

Digital design and computer architecture, or DDCA for short, deals with the essence of computers at a lower abstraction level. This ranges from simple logic gates all the way to branch prediction and memory optimization.

Parallel Programming, or Pprog for short, discusses, as the name suggests, the idea of parallel programming. Starting off with manually starting threads over using ForkJoin pools to transactional memory and STM. We looked at the benefits, but mainly the flaws of each of the implemented methods.

During this period of “relaxation” I will only be posting in this format. Hopefully there will be a few interesting topics for you too :)

]]>This will be a personal work where I outline interesting projects, small guides and other little quirks. The content will be mainly based around CS-related subjects as well as some interesting mathematical concepts.

As any other blog this one has three components: The home page, where you will get a rough overview of the blog, the about page, where you get to find out more about this blog as well as about me and last but not least the `Posts`

page where all of my posts will be listed.

Polyring is a webring made by JulianXY. This webring links to like-minded CS students that are currently studying at ETH too. By clicking `next`

you get to the next blog in the ring, where you’ll find yet another polyring. This is a great way to discover new high-quality blogs.

To finish this up, I hope that you, whoever decides to visit this, have a good time and if yout noticed something unsettling or want to contact me, you’ll find my details on the homepage. Thanks for dropping by :)

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