In the 121 weeks I’ve been working through KA, I’ve never gone through a week like this one. I got my ass handed to me this week trying to get through the last exercise of Integrals and pretty much made 0 progress towards getting through the exercise… Let me give you an analogy of what happened… 

I view what I’ve been doing as walking up flights of stairs. I started at the ground level (arithmetic) and walked up the first flight of stairs to the the second floor (algebra 1). I couldn’t have ‘jumped’ from the ground level to the second floor but by taking it one stair at a time (i.e. one section/video/exercise at a time) it wasn’t too difficult to make my way up to algebra 1. On my journey up these flights of stairs, every once in awhile I would come up to a stair that seemed MUCH bigger and more difficult to get over than the other stairs. (The unit circle comes to mind.) Sometimes these stairs were bigger because the concept I was learning was just simply a more difficult thing to understand than the other concepts I’d learned. Other times, however, I felt like the reason the stair was bigger was because KA hadn’t properly prepared me for it and that there were a few missing stairs that should have been infant of it. That’s how I feel about the exercise I worked though this week. 😠

(That being said, I’m well aware that I’ve been using KA for free for the past 121 weeks and I’m that I’m coming off as incredibly entitled and ungrateful which I’m not! I was just frustrated this week.)

As I went through the exercise Integrating Using Trigonometric identities, what was the most frustrating part was that I had no intuition for which trig identity I should be using. The section had 3 videos in it that I watched before working through the exercise and most of the videos used a simple variation of the identity sin²(x) + cos²(x) = 1 where I’d replace a sin²(x) with a (1 – cos²(x)). When I was given a question in the exercise where I needed to use this identity, I was usually able to solve it. For example:

The exercise had a number of questions in it that used what seemed like more advanced trig identities which I didn’t understand very well. These included double-angle identities, half-angle identities, reciprocal identities, etc. On every one of these questions, I had no clue which trig identity substitution I should be trying to use. Eventually, I resorted to looking at the picture below and guessing which identity I should try and use, but it never worked.

What was also incredibly frustrating was that, even once I got the question wrong and was shown how the integral was supposed to be evaluated, I could barely keep up with the algebra. 😔 The solutions I was shown often skipped steps and were hard for me to keep up with even if they didn’t. Here’s an example of a question I got correct but took a guess on and didn’t actually understand what was going on:

What I didn’t realize in this question was that (1 – sin²(x)) is equal to (1 – sin(x))(1 – sin(x)). I thought that it would have been equal to (1 – sin(x)sin(x)). This was just the tip of the iceberg when it came to trig algebra that I didn’t understand in this exercise. Suffice it to say, by the middle of the week I was very demoralized and felt like I was making no progress.

I hit a breaking point when I was given this question:

(Even though it says I got the question correct, I didn’t. I just clicked on one of the answers and I happened to click on the correct answer.)

When I looked at the solution, I had absolutely no clue what was going on. In step 1/4, it states to use “integration by parts” and goes on to label part of the functions inside the integrand as u, v, du, and dv. Part 2/4 then moves sec(x)tan(x) outside the integrand. After reading that I was ready to scream because I had no idea what was going on. I Googled “integration by parts” and realized that I had never been shown how to use this technique. (This is what I’m talking about when I said I feel like KA didn’t prepare me for this section.)

I watched a few random videos I found on Google about integration by parts and eventually found this video which really helped:

I only watched the first ~30 mins of this vid where the prof goes through the formula and a few examples. I copied out the formula and the first 2 example questions into my notes:

            Question 1:

• ∫ x * e^x dx = ∫ u * dv
  • a(x) = u = x
  • a’(x) = du = 1 * dx
  • b(x) = ∫ e^x dx
    • e^x
  • b’(x) = ???
• ∫ u * dv = u*v – ∫ v * du
  • = x * e^x – ∫ e^x * dx
  • = x * e^x – e^x
  • = e^x (x – 1) + C

Question 2:

• ∫ x³ * ln(x) dx = ∫ u * dv
  • a(x) = u = ln(x)
  • a’(x) = du = 1/x * dx
  • b(x) = v = ∫ x³ dx
    • = x⁴/4
  • b'(x) = ???
    • (Note: I’m realizing as I type this that I REALLY don’t understand what’s going on here. In my mind b’(x) should be equal to x³ and b(x) should be equal to x⁴/4.)
• ∫ u * dv = u*v – ∫ v * du
  • = ln(x) * x⁴/4 – ∫ x⁴/4 * 1/x dx
  • = ln(x) * x⁴/4 – ∫ x³/4 dx
  • = ln(x) * x⁴/4 – 1/4 * ∫ x³ dx
  • = ln(x) * x⁴/4 – 1/4 * x⁴/4
  • = x⁴/4 * (ln(x) – 1/4) + C

After going through this video and the two example questions, I Googled “KA integration by parts” and realized that it’s taught in Calculus 2. Where it’s shown in the KA Calc 2 section, there’s also a bunch of videos about trig substitution in a section just before it:

I was a bit annoyed that these 2 sections weren’t given to me already, but I’m happy that I was able to find them and am hoping they’ll help me understand both trig substitution and integration by parts. My plan of attack for the next week or two is to go through both of these sections and then come back to the trig sub exercise that stumped me this week once I finish both sections.

I clearly won’t make it through Integrals (2,530/3,200 M.P.) by the start of the New Year which is disappointing. Nonetheless, considering how demoralizing the past week and a half has been, I’m happy to have a new plan of attack to try to figure out how these things and integrals, in general, work. And even though I won’t make it through Integrals by the New Year, one milestone I literally just reached is having written 150K words of rough notes for this blog!

WOO!! 🤓 🎊