The Chain Rule Reversal

We started the class, reviewing the format we should post our blogs in. Key points would be white space (Paragraphs), because the brain digests it easier instead of looking at it like a huge beast. Another would be to post pictures in the centre format, so that it's separated from the text and easier to see.

Moving along, we started the lesson with different ways to antidifferentiate, for example, breaking up the terms so that it's easier to antidifferentiate, as well as algebraically massaging the expression into a form that it's easier to antidifferentiate. Speaking of massages, I could use one right now thanks to all that working out, making my muscles sore. Here, we separate the terms, and move the constants (2 and 3) outside of the integral. And then we can antidifferentiate sinx and e^x easily, and after antidifferentiating it, it becomes...
Note: *Don't forget the + C. It was evident during the class that we constantly forgot to add the + C. It costs a mark!*

Now, looking at the slides, it didn't turn out so well because the transparency effect didn't work so you can't really see whats there, but I'll spell it out for you.

So we had the integral of the function (x^2 + 1)/x on the interval from 1 to e with respect to x. Because we don't know how to antidifferentiate quotients, we can massage that to separate terms so that they're both divided by x. The x's reduce on the first term and that leaves us with the integral of the function x + 1/x on the interval from 1 to e with respect to x. From here, we can antidifferentiate this, a LOT easier than before.
We now have x^2 /2 + lnx evaluated on the interval from 1 to e. Now we get...

After evaluating, we get the final answer of e^ + 1 / 2. This leads us to conclude to not differentiate too fast, and algebraically change it so that it's easier to differentiate it.

But on to the main dish. If we can differentiate multiplying terms, can we antidifferentiate multiplying terms as well? Well we tried that out with this question. Mr. K pretty much confused us when we asked if there was a product rule for antidifferentiating, and his answer was a "kind of, not really, definitely, maybe" answer. So, basically, yes, but mostly no. -__-"

As you can clearly see, it didn't work out so well when we tried multiplying and antidifferentiating. We even separated the terms! However, there is a product rule for antidifferentiating.

We were introduced to the Product Rule. Now, we weren't exactly sure if this would work, but it works like this. The integral sign cancels the derivative sign, so we're left with the Integral of the derivative of the outer function at the inner function multiplied by the derivative of the inner function, which should equal the parent function. So then we put it to the test.

Yay, it works. The F'(x) = cosx. g(x) = 3x and g'(x) = 3. So working that out, we get sin(3x) + C. That's right, + C and don't forget it. Always makes sure to analyze to see if they're composite functions. But that led us to the thought, what if that 3 wasn't there? How would we know how to do it? Well we tried just that too.

So because there's no g'(x), we multiply by one using a full fraction using the coefficient number from g(x). From the fraction, remove the half from the integral so that we're left with F'(g(x)) * g'(x), which is sin(2x) * 1/2 + C <-- There it is again! OMG!

Anyways, that wasn't fun because I fell asleep for an hour and woke up really tired, so I'm happy that I'm finished, 'cause I have to study for Chemistry now.. Like, now. I forgot, I have to crown the next scribe. Hi, I'm Justus. No, I'm not really Justus, I'm saying that Justus is the next scribe for the last day of the semester. Have fun guy.