Techniques and Substitution

So, I know this is long overdue, and I'm sorry guys I've been ridiculous busy. Anyways it's not monday yet, so technically, it's not late but yeah ;p

Anyways here we go with the blog for jan 23rds class on substitution! :]

Alrighty, so in our quest to be able to anti differentiate effectively we've found that its much easier to go forwards, then it is to go backwards, because the nature of the original problem isn't always apparent from the differentiated version (ie, the result, isn't always indicative of what the beginning was like.)



Now the whole idea of substitution kind of ties into running the chain rule in reverse thing (as seen in the above image.) In the example function to the right of the definition box thinger, the inner function is shown to be

x² + 3.

Well using substitution we just change this out for something slightly simpler. Sayyy, u? Sounds good to me. So now that we've got that out of the way, we need to actually differentiate the value of u (which we know, by the instructional box, to be g(x). In this example, we end up with

u = x² + 3
du = 2x dx

Now, what we want to do, is isolate xdx. To do this, we divide du by 2

du/2 = xdx

So after thats said and done, we end up with





Which can then be converted into





THUS we are on the next step in the "down, over up" process, that being anti differentiating integral symbol F'(u) du (aka, anti differentiate what we have above.) Doing this gets us




Continuing along with the process, we now have to substitute the value of u back in, in order to get our final answer.



And we're done! Now that might not seem like to much to do, but it's VERY VERY important to follow the process, and have good bookkeeping. IE. Keep track of your substitutions, and where everything is, because it's very easy to get lost and things.

Now on the next slide, is another example of substitution in action.



However, what in these two examples, all the "pieces" are these so to speak. But, what do you do, if they aren't? D:



In the example above, you see that if x is equal to -7, some problems arise. However, as seen in the slide, we may get around this problem via some algebraic massage.

Since u = x + 7

Then you can say that

u - 7 = x

and

du = dx

From there its all a matter of following the process like we have been for every other question, albeit, with slightly more algebraic massage :D

Alright guys, I think that about sums everything up. Hopefully it all made sense D:

The next scribe shall beee (checks list) J + Me, since I've hooked her up with a bunch of new tunes, she needs a good excuse to sit on her laptop listening to them, and a scribe post is just that excuse I think! :]

Now I'm off, to finish our video D:

Ciao!