Integration and Logarithm questions CIE AS Mathematics

You need to show us what you have done.

I'll give you some clues:

1) Do you know a trigonometric identity involving tan squared?

2) Let y = x^1/3 or cube both sides?

Well,

1. According to my formula sheet, tan^2 = sec^2 -1 , and the integral of sec^2 = tan + c. I am not really that good with integration when trigonometry comes in the picture, so im guessing here: using this identity, does it mean that the integral of tan^2x = sec^2 2x - 1 and therefore sec^2 (2x) - x | ?/8, 0 ?

2. I guess that would help

So the integral of sec^2 (2x-1) dx = tan (2x-1) / 2 + k ?

I'm sorry if I'm wrong I'm a little confused right now

Because we said that due to the trigonometric identity, tan^2 x was the same as sec ^2 x -1

$\tan^2 x = (\sec^2 x) - 1$

See the difference?

Aha, I see. So we can say that the integral tan^2 2x is the same as the integral of sec^2 (2x) -1 ?

Yes so use the rule I gave you (ignore the + b bit if it is confusing you).

So with your rule does it become [tan (2x) - 1] / 2 + k

You are really struggling here because I think you are trying to run before you can walk. How are you at basic integration? Do you understand that:

$\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx$

You may be right and I appreciate that you are trying to help me. To answer your question, yes, I understand that. I'm not sure what my problem is, but I can tell you that, for example, I can easily integrate something like x^3 + 2x^4 or something but when something like tan^2 2x or other trigonometric function comes into the picture I start to get confused as you can see.

Do you happen to know any resources which I can use to aid my understanding, or, do you know somewhere which clearly underlines the most important rules I need to know regarding integration? Or could you write some out for me? I would very much appreciate it as I am taking my CIE AS Level exams in a month and this is one of my weaker areas which I need to work on.

I can send you some Powerpoints on simple trig integration but they are designed for OCR A2 rather than CIE AS so may contain stuff you haven't met yet and could confuse you further.

Anyway back to the point, can you integrate this ...

$\int 1 \, dx$

as this is the skill you seem to be lacking.

Okay, as embaressing as this is, you are right, I saw my teacher doing that once or twice in class but I suppose I wasn't paying much attention.. But isn't the integral of 1 = x ?

Yes so an x should be appearing in the answer to this question too.

I see. Do you think the powerpoint presentation would help? I mean tbh I would like to get all the help I can, and I was thinking earlier today it might help to have all the rules I need to know written down somewhere... Let me know what you think. Any tips advice etc I would also very much appreciate

Thank you!

Pm me with an email address and I will send you some stuff later. Just going out now. Have you finished that question now?