## Quality Pictures

Just to reiterate my piece just now, here are the pictures of such people.

## Quality Time

Last Saturday, Shun and I went to Wild Wild Wet to chill and have fun. We paid $23.80 and another$8 for lunch to play what was essentially the old Wild Wild Wet. We exhausted all the rides in the first two hours then decided to slack at the Tsunami.

As we did, I reflected and shared with him how water theme parks in reality were essentially the same. Most have a lazy river, some thrilling rides, a mass pool where the waves come and go. It wasn’t the attraction that made the day great, but the company. It’s usually not the activity in and of itself that’s great, but the people you spend time with in that activity.

I remember watching Trolls and the movie in and of itself was meh. The movie was quite adorable but the story was somewhat cliché. Yet, it wasn’t the movie that made the day great, but the company.

I remember catching up with friends over food. Some bring me to Nando’s for an exquisite taste (aka daylight robbery) while others tight-budgeted friends like me brought me to McDonald’s and others to Makisan. Yes, in all cases, the experience was equally great, not for the food or the cheapness, but because I got to catch up with my loved ones.

I remember heading to Christmas Wonderland in the Gardens, not because the decorations and the music were good, though I did fall in love with Pentatonix’s rendition of ‘O Come, All Ye Faithful’, but because the close friend who went with me there did so out of compassion.

I remember each cycling session, Pulau Ubin, Biking in the Rain and ECP to Pasir Ris, not because they were epic misadventures but because we went through them together.

I remember studying for hours on end, not because doing work was awesome in and of itself, though I did enjoy preparing my Math worksheets, but because I got to do them with the people I love and trust.

I remember every SFC session was amazing not because of the planning per se but the unity of the CCA amidst doing so.

I remember jamming not because we were fantastic musicians but because we in our awful voices and skills (or lack thereof) still chose to enjoy the time together.

I remember SUPER FUN not because I was able or unable to teach but because the people I taught were incredibly teachable.

In reality, it’s never the activity itself that I’m happy about, though fun activities do attract us more, but rather the different company with whom I engage these activities in that produces a smile to my face.

And no matter how under- or over-priced these activities were, I can with full confidence say that it’s all worth it.

Thank you all who chose to invest in my life. I cherish every moment of it. Cheers to many more moments like these to come!

## Self-centeredness and its Fruit

“For men will be lovers of themselves, lovers of money, boasters, proud, blasphemers, disobedient to parents, unthankful, unholy, unloving, unforgiving, slanderers, without self-control, brutal, despisers of good, traitors, headstrong, haughty, lovers of pleasure rather than lovers of God, having a form of godliness but denying its power. And from such people turn away!” [2 Timothy‬ ‭3:2-5‬]

Last week, the Lord prompted in my heart, “notice what is the first of the sins,” pretty much confirming all that I shared about self-centeredness as the root of all evils. Loving myself here does not mean thanking God for who I am, but rather looking at myself so highly that I don’t need God. And in reality, we all fall into this sin every once in a while with our lives. Is there grace for forgiveness and restoration? A million times, yes! Yet, God designed in such a way that it’s healthy for us to confess our sins to quicken restoration, and in light of that, I’d like to share three ways this has been apparent in my life.

1. Jealousy. One reason I quit Instagram is to not feed myself triggers of people’s adventures. Yet, it struck me that I’m the only one among my closer group of friends who isn’t travelling overseas this year. It kinda dawned upon me that life stank, especially since even my free time was not used productively in catching up with people.
2. Self-pity. This is a result of (1) but yeah in response I kind of started to wallow in self-pity, in questioning the meaning of my life and how compared to others it isn’t even a life.
3. Emotional attachment. Also, my happiness was greatly dependent on whether I spent the weekend with my friends or not. If I did, I’ll be happy, if not, then I’ll be sad. My joy didn’t come from God, and therefore would be easily taken away if I don’t get what I want.

These were some thoughts the Lord brought to my attention last night as I prayed. In response I repented (changed my mind) on these issues. What makes me think that just because I remain in SG that it has to be fruitless? What makes me think that just because my life revolves about God, Math and Music, it means that I must be a boring prick? What makes me think life is meaningless without endlessly busy meetups?

There are people living lives that describe what I share but are either fulfilling or not. They are fulfilling when Christ is the centre of their lives and not when their reasons correspond with those in my struggles.

I guess for today I’d like you, dear reader, to know that in light of my arguments and my stands on many issues, I’m an imperfect human who messes up from time to time, and these are things I struggle at and will appreciate your prayers in.

## By Parts Backstory

Here’s the simple concept behind integration by parts as described in an earlier post. Integration by Parts simply what I call the reverse-product rule. Here’s why.

Let $u$ and $v$ be functions of $x$. Using product rule, when we differentiate $uv$, we differentiate the first term *times* keep the second constant *plus* differentiate the second term *times* keep the first constant. That is,

$\displaystyle \frac{d}{dx} \left(uv\right)=\frac{du}{dx} v + u \frac{dv}{dx}$.

Notationally, since $u'=\displaystyle\frac{du}{dx}$ and $v'=\displaystyle\frac{dv}{dx}$, we rewrite the above identity as

$\displaystyle \frac{d}{dx} \left(uv\right)=u'v + uv'$.

Let’s integrate with respect to $x$ on both sides. On the LHS, we are integrating what we get after differentiating. Since they cancel each other out, we get $uv$ on the LHS. On the RHS, we get the integrals of each chunk added together, that is,

$\displaystyle uv = \int u'v\ dx + \int uv'\ dx$.

Subtracting by $\displaystyle \int u'v\ dx$ on both sides and switching the LHS with the RHS, we get

$\displaystyle \int uv' \ dx = uv - \int u'v\ dx$.

which is the famous integration by parts formula.

Hope this was insightful on how this technique is nothing more than reversing the product rule!

## The Self-Pity Ends Today

All the free Sunday afternoons that end up spent lying on my bed questioning about life.

All the half days I get in Friday afternoons generally not well spent unless I make the first move.

All the time wasted throughout the day scrolling and refreshing through Instagram and not being productive.

It ends today.

From today on, I’m choosing to enjoy each break I get. I’ll enjoy each weekend to the fullest either with people or on my own. With the new task of preparing crash course material for O Levels, which include mainly algebra and functions and drills from other topics, one can see how easily I can spend my time. I’ve already quit Instagram since a few weeks back, am going great without it, and when I do return to it, plan to use it once per week only.

And I’ll stop comparing my life with others. Others might have more time supposedly well spent overseas, dramas, meetups and whatnots. Well, my life doesn’t have that many glitters, but that doesn’t mean I need to wallow in self-pity. My Instagram account may not be as active, but that doesn’t mean life needs to be uneventful. No. That false belief ends today.

I’m going to love myself because God first loved me, and I’m going to enjoy my weekends regardless of whether my loved ones are available or not. My life revolves around Christ first, then them. If they wanna meet during my unplanned free time (which my default means doing Math) then sure, I’m more than glad! 😁 BUT I don’t want my happiness to depend on my meetups with friends because I don’t wanna sulk at the lack thereof otherwise. I don’t want to be dependent on the likes I get on my Instagram stories but want to depend on the Lord’s story of my life. I wanna depend on the Lord for my joy.

I choose today to love myself and seize every opportunity either to hang with friends or do Math. My principles in cherishing one another extends to cherishing myself and the person God has made me to be. My meetups won’t end, but the dependence on them for my joy will. They end now. The self pity ends now.

—Joel Kindiak, 6 Dec 2016, 0013H

## Integrate by Parts, the IS-ID Way

In calculus, we learn to integrate products like $x \ln{x}$ and $e^x \sin{x}$ using the technique integration by parts. We use the formula,

$\displaystyle \int uv' \, dx = uv - \int u'v \, dx$,

and using LIATE (Log-InvTrig-Algebraic-Trig-Exp), choose the right-leaning term as $v'$ (e.g. $x$) and the left-leaning term  as $u$ (e.g. $\ln{x}$). We integrate and differentiate respectively to get $v=\displaystyle \frac{x^2}{2}$ and $u'=\displaystyle \frac{1}{x}$ respectively, substitute into the formula, simplify the expression and carry on our calculations. Rather than confuse you with which expression to use as $\displaystyle v'$ and $\displaystyle u$, I propose a simpler presentation.

Before we discover this presentation, I want to clarify that this is just a simpler presentation of the same technique. The concept is the same. The presentation simply makes more intuitive sense. Also, I might derive the formula in another post. Integration by parts is essentially what I call the “reverse product rule”, and I might elaborate more next time. For now, lets take a look at the formula one more time, but this time I highlighted in green $v$ and $v'$ and red $u$ and $u'$, and swapped the order of multiplying (doesn’t change the formula mathematically):

$\displaystyle \int$ $v' \,$ $u$ $dx =$ $v$ $u$  $- \displaystyle \int$ $v \,$ $u'$ $dx$.

Notice that $v' \,$ gets Integrated and $u$ remains the Same in the first term, $v$ $u$. Notice that $v' \,$ gets Integrated and $u$ gets Differentiated in the second term, $\displaystyle \int$ $v \,$ $u'$ $dx$. Thus, we get this idea of Integrate * Same – Integrate * Differentiate, or succinctly put,

$I$ $S$  $- \displaystyle \int$ $I \,$ $D$ $dx$.

As mentioned, this is essentially the same method, but using these letters makes it easier for us to choose which terms to use. Let’s integrate $x \ln{x}$ as an example.

Between $x$ and $\ln{x}$, the right-leaning term is $x$. Thus, we Integrate that to get $\displaystyle \frac{x^2}{2}$. The left-leaning term is $\ln{x}$. Thus, we Differentiate that to get $\displaystyle \frac{1}{x}$.

Using IS-ID, we plug in the relevant letters to get

$\displaystyle \int$ $x \,$ $\ln{x}$ $dx =$ $\displaystyle \frac{x^2}{2}$ $\ln{x}$  $- \displaystyle \int$ $\displaystyle \frac{x^2}{2}$ $\displaystyle \frac{1}{x}$ $dx$ $\displaystyle = \frac{x^2}{2}\ln{x} - \frac{1}{2}\int x \, dx$.

Integrating the remaining portion, we get our final result,

$\displaystyle \int x \ln{x}\, dx = \frac{x^2}{2}\ln{x} - \frac{x^2}{4} + C$.

I hope this alternative presentation will help you more effectively and integrate by parts with better understanding. As an exercise to the reader, find $\displaystyle \int e^x \sin{x}\, dx$. The final answer, for your reference, is $\displaystyle \frac{e^x}{2} (\sin{x} - \cos{x})$ and you get bonus marks if you can explain how this answer can be improved.