5.6.2

[STARTS AT 2:42]

INSTRUCTOR: Number 11 looks a lot more complicated. It really isn't that crazy because we want this to look like 1/x, right? Well, if I make my u equal to-- I'm going to go ahead and rewrite this. 2x cubed plus 3x over x to the fourth plus 3x squared. The wise idea here would be to make u equal to x to the fourth plus 3x squared. The reason there is that when I take the derivative of that, I might get something with a couple of lower exponents, which is what I happen to have in my numerator. 

So oftentimes, when I see something like this, making the denominator equal to u, if you take the derivative and it becomes something that looks like your numerator, that's a good idea. That helps. So this would be 4x cubed plus 6x dx. Now, that does not look like what I have in my numerator. 

However, it differs by a factor of 2. So if I divide everything by 2 here, or in essence, make this 1/2 du, that is 2x cubed plus 3x dx, which is what I want. So this will turn into the integral 1/2 du, and it'll be 1/u. So I have 1/2 natural log of u x to the fourth plus 3x squared plus C. That's really all there is to that. Even though it looks pretty crazy, that's all there is to it.

[ENDS AT 4:30]