This week I got a stronger intuitive understanding for how to go ‘back-and-forth’ between an antiderivative and it’s derivative, but I can tell my understanding of it has still barely scrapped the surface. To be completely honest, I still don’t even really understand what the difference between an integral and an antiderivative is. (I just tried Googling it and, after reading a few articles, I still don’t really understand. 😔) Even though I have a long way to go before I figure out integrals, I’m happy with the progress I made this week and feel like I took some solid steps in a good direction. Oddly enough, a lot of my week was spent solving the derivatives of trigonometric functions which then helped me understand and figure out how to find their antiderivatives. This helped make antiderivatives a lot more clear and it all started to come full circle! (No unit-circle pun intended.)

I began the week going through the final article in the section Reverse Power Rule. After spending most of last week confused about how this rule works, for some reason it clicked for me on Tuesday and, not only did the entire article seem clear and obvious, but I also got through 4 exercises on the reverse power on my first attempt. (💪🏼) This rule states that when you have an expression in an integral that has a power on it, you can integrate it by adding one to the power and dividing the expression by the new power. (Side note – I’m not sure if I’m even using the terms ‘integral’ and ‘integrate’ correctly here.) Here’s an example question:

• ∫4t^1/3dt = ((4t^{1/3 + 1})/(1/3 + 1)) + C
  • = (4t^4/3/(4/3)) + C
  • = ((4t^4/3) * 3/4) + C
  • = 3t^4/3 + C

(Side note – I know that C stands for “constant” but I don’t really understand how/why it works or its purpose. I know that you always add it to the end of the integrated expression, however.)

After getting through the section on the reverse power rule, I spent the rest of the week working through a section called Indefinite Integrals of Common Functions. The first video I watched showed me what the indefinite integral of 1/x is. Since the derivative of ln(x) = 1/x, I learned that the integral of 1/x is ln(x), however there is a condition on x. I don’t intuitively understand the following, but if you graph the function of ln(x) the function will only have values on the positive side of the x-axis. Here’s a photo of the graph from Desmos:

On the positive side of the x-axis, the green function g(x) = ln(x) shows the values for ln(x) and the function f(x) = 1/x shows the values for 1/x which is g(x)’s derivative and, as far as I understand, g(x) is f(x)’s antiderivative. You can see that g(x) stays in the positive side of the domain whereas f(x) also has values in the negative side of the domain. In order for g(x) to represent the f(x)’s antiderivative across the entire domain, you have to add absolute-value brackets around x to give you g(x) = ln(|x|):

Here’s a page from my notes where I went over this concept:

As you can see from my note, ln(|x|) is the antiderivative of 1/x but there’s one condition, that x ≠ 0 since the function is undefined at that point. Going through this video helped me understand that the derivative and antiderivative of expressions are the opposites of each other (or maybe inverses of each other? 🤔) and can be used to go back and forth between the two.

Apart from learning that the antiderivative of e^x is e^x (which seemed somewhat obvious to me since the derivative of e^x is e^x), the rest of the section and the rest of my week was spent working on integrating trigonometric functions. Here’s a photo from my notes that shows the integrated functions of common trigonometric integrals:

Working through these questions on integrated trig expressions also helped cement the concept in my mind that finding the antiderivative of an integral is just the inverse operation of finding the derivative of a function. (Again, I’m not sure if that’s the right way to phrase it or even if that’s actually correct. 😵‍💫)

Here’s an example of a question I worked through:

As you can see from the question, all of the solutions to these questions simply stated something like, “since we know that g(x) is the derivative of f(x), we know that the integral of g(x) is f(x).” I found it really annoying that the solutions didn’t give any information or clarity on the math behind finding the integral. The way I ended up solving these questions was by taking a guess at which of the four answers I thought was correct and then, going backwards, found the derivative of the answer I chose and if the derivative of the answer was the same expression as in the integral I knew I had picked the correct answer. Here’s my work from the question above:

Here are two more questions I worked through where I used the same strategy to determine the correct answer:

As you can see from my work, not only did I get practice understanding the relationship between integrals and their respective integrated expressions, I also got a lot of practice finding the derivatives of trigonometric expressions. Working through these questions, I also FINALLY figured out how/why the product rule and the quotient rule are essentially the same thing when finding the derivative or two functions being multiplied and divided by each other. (The thing that made it click was realizing that when turning the quotient rule into the product rule, i.e. bringing the denominator up up into the numerator and multiplying them together, you need to switch the sign of the power from negative to positive on the expression being brought up and, in doing so, you then need to use the chain rule on that expression.)

All in all, I’m happy with the work I got through this week and what I was able to learn. I managed to get 17% further through Integrals (1,840/3,200 M.P.) from the start of the week to the end of the week. It looks like I have a pretty good shot at getting through the entire unit before the start of the New Year which I’m still gunning for. As I mentioned at the start, I of course still have a long way to go before I have a full understanding of integrals, but I’m fairly confident that I’ll be able to figure it all out with a few more weeks of practice. 👍🏻