Category Archives: Math

Antiderivative of tan x

I was toying with Problem 317 on Project Euler when I found a messy (but rather easy) antiderivative of a polynomial (with messy coefficients of course). Somehow whatever train of thoughts led me to think of the antiderivative of tan x. (That’s extremely sidetracked, but whatever. My initial guess of 2.4 million something was wrong anyway 😛 )

And it is pretty surprising.

Letting u = \cos x, we have du = -\sin x \,dx. Hence
\displaystyle \int \tan x \,dx = \int - \dfrac{-\sin x \,dx}{\cos x} = \int -\dfrac{du}{u} = - \ln |u| + C = - \ln |\cos x| + C.

Wild Logarithm appeared! But seriously. This is pretty amusing. Well, if you know \dfrac{d}{dx} \arctan x = \dfrac{1}{1 + x^2}, this is not that amusing, but still.

Math has always mesmerized me with its beautiful, unexpected results. However, it requires deep thought to actually understand it, hence a boatload of dislikes to math. In my class of 22 students, only one goes deep to math (namely me); one or two are good at school math, a few more okay, but that’s it; the rest are…uh…terrible. Of course, you cannot ask the opinion of a person who really digs math about math in these settings (aka the question about how his classmates are progressing in math); that’s biased.

Recently, namely 2.5 hours ago, I’ve just finished tutoring seven of my classmates for about 5.5 hours about math. Tomorrow is math (and history) finals exam, and it involves four topics: antiderivatives and integrals, matrices, vectors, and geometric transformation. I almost gave a very terrible problem (find the area enclosed between y = e^x - 1 and y = \ln (x+1), later revised to be base 2 instead of e); it’s terrible just because finding the intersection points will be a nightmare for them. Although I’m positive that most readers of this post can solve the problem without too much effort (note that the former is convex and the latter is concave, so they cut at at most two points, and they can be found easily).

Appreciating math (or any subject for that matter) is hard. I tend not to appreciate Indonesian (mostly because of severely flawed and broken “opinion-type” questions, where English should have anyway but our English focus more on grammar and reading); there might be someone out there who loves Indonesian more than their spouse (if there is any). Sometimes someone simply doesn’t appreciate any subject at all. I don’t have any right to judge; it might be that they like something else.

However, if you have the patience and logic to grasp the hidden beauty of math masked by mindless exercises such as \int x \sin x + x \sin x^2 \,dx, you might just stare in awe (or in confusion; again note that you need logic) of the beauty of math. You might even be amazed of the wonders of antiderivatives; note that the above example employs both partial integration (for first term) and integration by substitution (for second term) while differing only a number in its expression. So beautiful, so orderly.

Now that I’ve written too much, I might better sleep and prepare for tomorrow. It’s going to be a big day. (Those seven will come again for tutor on Tuesday’s Physics. 😛 )

[IMO 2012 Post 5] Games and More Games

And now we’re back from a 12-post series of puzzles to the silently postponed IMO 2012 series.

In a nutshell: It’s too dark. Uh well I can’t summarize this neatly, but let’s say we have a bunch of games and stuffs.

Fun diversion: The linked picture (hopefully you can see it; it’s in IMO 2012’s Facebook page) has the Indonesian team at the lower-right corner. Guess who is which. Participants names can be seen in this page.

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[IMO 2012 Post 4] Six Problems

Hm I start to forget stuffs in IMO. Better get these out before I forget it entirely.

In short, I did extremely poorly in it, at least for my target of getting 21 points or silver whichever is higher. Stuffs aside, I’m having moar fun in the recreation hall, especially after Day 2’s contest where the stuffs of recreation hall becomes available.

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[IMO 2012 Part 3] Recreation Hall

Hm for an absolutely unrelated opening text: I find out the following comment from someone, filtered in my Spam. I have a hard time deciding whether it’s spam or not -_-

“I went over this internet site and I conceive you have a lot of great info, saved to my bookmarks (:.”

IMO 2012, Post 3. In short, I find the recreation hall. Stuffs basically don’t deviate far from that thing.

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[IMO 2012 Part 2] A Long Journey

Too much Where’s My Water?. Also still a bit jetlagged.

In short, the journey to finally reach the official hotel (NH Gran Hotel Provincial) is long. At least for Indonesia team. But we made it past flights totaling over 24 hours, extremely cold temperatures for us tropic people, and…err…stuffs that are unusual.

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[IMO 2012 Post 1] Summarizing

Okay I’m a bad blogger for not posting for 1.5 months, but that’s another story.

So, as you might have known, I was going to IMO 2012 in Mar Del Plata, Argentina. If you know my real name, you know that I’m one of the “unlucky” ones of getting a double-digit score and not getting Honorable Mention. Or at least I consider those people unlucky, myself included.

I will begin (or try to) a series of posts retelling how my trip to the opposite side of the world was. (If you don’t get it, I’m in Indonesia and I went to Argentina. Yeah, opposite.) Hopefully. I’m still having jet lag and is now in a half-conscious state, being able to fall asleep at any time.

Warning: This post contains a lot of math stuffs. Read at your own risk.

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Goals

UPDATE (20 Nov 2012, 19.46 UTC+7): Okay this is obvious

Math, career, and life:
– IMO participant (done) Gold IMO medallist (IMO 2012 fail. Now 7m)
– Get scholarship for college (7m)
– Has own website domain and a well-known internet company (15y)

Programming:
– Fluent in C++ or Pascal or Python or Java (2m)
– Solve 250 Project Euler problems (7m)
– Get URLQuest 3 well-known (2y)

Games:
– Catch all legends in HG (done)
– Get BW2 (done)

Yay. I have bad life goals.

Long time no post

And hence I should let my readers to know that this blog is not dead.

Things I learn:

– Double negative (2C – 2D!, 2H – 3C!) is bad for games. 3H is down 1, although we hold like 8 or 9 hearts (I forgot) and stoppers in two side suits. Yeah, only two side suits. The other has two losers before I get it void.

– In Virtual Villagers 5: New Believers, HEATHENS CAN DIE. I’m not sure where my Heathen Master Scientist went, but it certainly made me unable to complete the puzzle involving them and another involving the Chief. Which means no earthquakes. Ugh, I don’t even get revive. No fish + 43 people = 0 food. Currently restarting.

– In osu!, streams are hell. In Pachelbel’s Canon (Funtastic Power!), Canon difficulty, I failed at the last 5 seconds of the song because apparently I sped up my rhythm during the last stream, missing the later notes and eventually the attempt. 😦

– Making sufficiently hard geo problems is hard 😦 Making sufficiently hard combo is easy though 😀 (written by the mind of a combo-lover-and-geo-hater olympiad math person)

– I can’t make good room designs 😦 This means I will rarely post Surveyors Heyawake. I’m absolutely sure it has more tricks than what I have in Puzzle 4 though.

– Yay. What do you call it if you have a memory card filled with DS games? Whatever it is, here’s a story. I have two DS, and I also have a MicroSD-to-computer “hub” or something. So, I copied my save data of Pokemon Platinum to computer (and also backed up the contents of one memory card), then do a trade (well, eight trades to be precise) between my Pt and HG. Later, after I finished moving all I want to HG, I’ll see the result of replacing back the back up as my actual Pt data. So, cloning in a new form perhaps? The Pokemon that will disappear along with the temporary Pt are worthless ones. Oh hey I haven’t caught a Magikarp in HG.

– Also in conjunction with above, I used whatever the “cheat-by-toggling-which-cheats-you-want-to-use-when-you-load-the-game-by-the-OS-(or-whatever)” to get the three event items in Pt (Member Card, Oak’s Letter, Azure Flute). I caught Darkrai and Shaymin, but Arceus is Lv80 and my team is only Lv74 at most 😦 Whatever. And hey Shaymin. Yeah I sent it to my HG too. Currently training.

– I decided to recruit people as “development team” of URLQuest 3. If you have ideas of what to make a good online riddle website, e-mail me. Did I write my e-mail in the About page?