Integrating 3cos[SUP]5[/SUP]3xdx

[bobsmith76]

Homework Statement

If you look at the answer you will see that they integrate sin⁴3x into (sin⁵3x)/5. well, if you can do that why not just integrate 3cos⁵3xdx into (cos⁶3xdx)/2? why go through all the trouble of dividing it into parts? it is true that (cos⁶3xdx)/2 = 0 but i still don't see why you can integrate sin⁴3x and not 3cos⁵3xdx.

 

[Mark44]

Homework Statement

If you look at the answer you will see that they integrate sin⁴3x into (sin⁵3x)/5.

No, they do not. They integrate sin⁴(3x)*cos(3x)*3dx[/color] to get (1/5)sin⁵(3x).

well, if you can do that why not just integrate 3cos⁵3xdx into (cos⁶3xdx)/2?

Because that's not the right answer. IOW, (cos⁶3xdx)/2 is not an antiderivative of 3cos⁵3x.

why go through all the trouble of dividing it into parts?

To make a relatively difficult problem amenable to solution by a very simple technique. Do you recognize what type of integration they did for the 2nd and 3rd integrals? I'm guessing you don't.

it is true that (cos⁶3xdx)/2 = 0

?
No, this is not generally true.

but i still don't see why you can integrate sin⁴3x and not 3cos⁵3xdx.

 

[Gallagher]

They don't integrate sin⁴3x into sin⁵3x/5 but rather, they integrate sin⁴3xcos3x into sin⁵3x/5.

Thinking about sin⁵3x/5 this makes sense as if you were to differentiate that function you would get nf^n-1 per the power rule which is sin⁴3x times the derivative of the inner function per the chain rule, cos3x.

You couldn't integrate 3cos⁵3xdx into (cos⁶3xdx)/2 as you would need a -sin3x with it so as to fullfil the chain rule.

I hope that I'm not totally wrong on this and if not, that it helps you understand.

 

[Mark44]

Bob, I have to question the reasonableness of your study strategy. I've read your blog, so I understand a little of your situation and what you're trying to do, which is laudable. However, mathematics is not something that is amenable to cramming X number of pages into Y hours of study time. You will not learn very well by merely looking at worked problems and trying to understand the steps they show. You'll learn better by actually working the problems - then things will stick in your mind.

Also, in a fair number of your posts, I've noticed a lack of understanding of basic algebra and trig concepts. You will continue beating your head against the wall, unless you revisit those parts of mathematics and reinforce these foundational subjects.

Unlike some other disciplines, mathematics is not made up of discrete, unrelated areas, but instead builds on previous areas. The old saying is, if you build your house on sand, it will collapse.