I had a more successful week this week than I did last week. I made it through the Integrals unit test on Thursday but got 5 of the 32 questions wrong. Considering I had gotten 3 of 8 wrong the week before, however, I was happy that I only missed 2 of the remaining questions. I’m still a bit bummed at how long it’s taking me to understand integrals. I literally started this unit on October 18^th which was 118 days ago, i.e. 17 weeks ago, i.e. 4.25 months ago… On a week to week basis, it feels like I’m barely making any progress but I know that in the big picture I’m slowly starting to understand how it all works. I also know that integrals are a key component of calculus so I’m trying to remind myself that it’s not surprising that it’s taking me this long. Still, I’m hoping I’ll finally get through this unit soon and begin to fully grasp how they work.

For some stupid reason, I didn’t screen shot the questions I got wrong this week. The following are questions that I managed to get correct (except for one of them) and was happy that I actually understood them without having to do any review:

This question stumped me for about 20 mins before I figured out a way to solve it. I tried to use some trig identities to break it up into other trig functions but couldn’t figure out how to do it. After a while, I realized I could just find the derivative of each of the answers and whichever answer’s derivative resulted in the function csc²(x), that would be the correct answer. I was happy that it ended up being the first of the four answers so I didn’t have to do every single one. I was annoyed, however, that when I looked at the given answer it it said, “We know that the derivative of cot(x) is –csc²(x)…” Maybe YOU knew that KA, but I didn’t know that… In any case, I was happy that I got it correct.

• ∫ 8x * (1 – x²)^1/2 dx = ∫ 8x * a’(b(x)) dx
  • (…)
    • a(x) = 2x^3/2/3
    • a’(x) = x^1/2
    • b(x) = 1 – x²
    • b’(x) = –2x
  • = ∫ (–4)( –2x) * (1 – x²)^1/2 dx
  • = (–4) ∫ b’(x) * a’(b(x)) dx
  • = –4 * a(b(x))
  • = –4 *2(1 – x²)^3/2/3
  • = –8(1 – x²)^3/2/3 + C

In the following question, it says I got it correct but in reality I had no idea what the answer was and completely guessed and just happened to get it correct…

The hardest part about this question is understanding the notation. I think I’ve started to wrap my head around what it’s saying in laymen’s terms (as you can see from my notes) but I’m still having a hard time understanding how the ‘index’ works. I find it confusing that the x on e^x turns into 3i/n. I’m pretty sure what that states is something along the lines of, “An ‘index’ of numbers is split between the x values 0 to 3 where the first number in the index starts at 1 and the index goes on to ∞. Each x value within the index (i.e. x_i) is equal to the index number times delta x, i.e. i * 3/n.”

The reason why delta x is equal to 3/n is because:

• Δx = “(Upper bound – lower bound)/number of terms”
  • = (3 – 0)/n
  • = 3/n

This is a question I straight up don’t understand. As you can see from the screen shot, the process to solve this question is incredibly simple. All you do is input x² in the place of t in the function within the integrand, remove the integral sign, and then evaluate the function. I have no idea why that’s the case however or why it works. I began working on an exercise that dealt with this concept on Saturday and realized it has to do with the Second Fundamental Theorem of calculus. I watched ~5 videos on it but still don’t understand how/why it works. 😔

Considering how many weeks I worked on understanding trig-substitutions, I was very happy that I eventually ended up getting this question correct. It took me 5 pages of notes and a bunch of trial and error before I got to the correct answer, however. I eventually realized I needed to split sec⁴(x) into sec²(x) * sec²(x). The problem I was having initially was that I thought I had to split the tan/sec combinations up to leave me with a tan(x)sec(x), i.e. the derivative of sec(x). One thing I was happy about was that I remembered the derivatives of both tan(x) and sec(x) and that I remembered sec²(x) = tan²(x) + 1 and tan²(x) = sec²(x) – 1.

Once I finished the unit test on Thursday, I started to work on the first of the 5 exercises I have to redo. I finished the first exercise before the end of the week which was called Riemann Sums in Summation Notation. In this exercise I reviewed how sigma notation works with Riemann Sums. It took me awhile to remember how it all works but eventually figured it out. I’m pretty sure this exercise was related to the third question in this post where I had to find the equivalent limit notation for ₀∫³ e^x dx.

This coming week I’m hoping to get through the 4 remaining exercises in Integrals (2,980/3,200 M.P.) by Thursday or Friday so I can restart the unit test before the end of the week. If I can manage that, there will definitely be a chance for me to finally get through Integrals in the following week which would be a HUGE relief. I don’t want to rush myself and cheat myself out of actually understanding integrals, but I do want to get through it and keep moving forward into calculus. I feel like my progress has been pretty slow lately and I’m getting somewhat demoralized because of it, so the sooner I can make it through this unit the better!