Notation question with regards to U Substitution for Integrating

[boldimage]

Hi there!

I was wondering if anyone could spare 2 minutes for what may be a pretty simple question with regards to u and du substitution for an integration problem.

Assuming we were using U-Sub and not just multiplying out the following function.

f(x) = 2(x)(x²+1)

We would let:

u = (x²+1)

du/dx = 2x therefore du=2x dx

So why when we rewrite the integral in terms of u and du i.e udu do you integrate the u and then sub in the x value and not integrate the du and sub in du?

I know from doing these questions that we add a 'c' (constant) and the du disappears but I don't understand why other than by convention. We have given it a value when we rearrange the du/dx above so shouldn't du be integrated and the 2x be subbed in also? When you integrate du by itself shouldn't we get u+c making the whole thing = u² /2 * u+ c

This may be really obvious but I can't make sense of it - any help explaining this would be greatly appreciated!

Thanks for your help guys!

M

 

[daon2]

I'm not sure what you are saying, maybe someone can better interpret. However, if you just recognize that you have a function times its derivative, it should be clear the indefinite integral is (function)^2/2 + constant. The "u's" are meant only as a tool to simplify how the integral looks so that you can apply one of the rules drilled into your head.

Also, to see why your answer does not make sense, take its derivative. Do you get back the integrand?

 

[boldimage]

Guys thankyou so much for your reply - I'm sorry Daon2 if I lost you there, I think my head has been in this textbook too long ha ha.

JeffM - you definitely captured the question as I understand it, and it definitely helped with the process but I don't know if I explained my confusion clearly.



I drew it on an image (hopefully it opens bigger when I post this otherwise i'll repost it below) to better explain myself. I know that the u's are used to rearrange a difficult expression but as we eventually replace them with what they originally stood for I was just wondering why we reinsert the x value/function for u and not reinsert the x value/function for du?

Hopefully that is clearer - if not I am meeting with a Professor tomorrow and will post what they answer with in case someone else is having the same confusion.

Cheers guys!

 

[cordoba]

Once you compute the integral and get (u^2)/2 + C, there is no du in the answer, so where would you do the subsitution?

But perhaps you are asking something more fundamental than this. What is your understanding of integration? What happens to symbol dx when you integrate with respect x i.e. \(\displaystyle \int x dx\) ? It's the same process regardless of the variable name (x or u). du is the differential with respect to u (instead of dx which is the differential with respect to x).

[By all means ask your teacher -- your question is a good one!]