Integration by substitution

[Skelly4444]

Having trouble with this integral. I have used the substitution as instructed by the book and obtain an answer which agrees with the book. However, when I try and use my own substitution, I obtain a slightly different answer and can't see why?

All my workings are attached; Method B is the book's method and method A is my own attempt.

Any advice would be appreciated.

 

[topsquark]

I'm not sure what the question is. When you integrate and backsubstitute in for x you get the same result for the integral. If the question is about the form of u being different in both integrations, remember that for a definite integral you would have two different sets of integration limits.

-Dan

 

[BigBeachBanana]

Both methods are valid. Finish the problem all the way through to confirm your answer. It might vary by the constant of integration, but it is still correct.

 

[Skelly4444]

Hi Dan,
Yes I can see that when I take the question through to the final answer, I do indeed get the same answer as the book using both methods.
I am puzzled though as to why the book chooses method B rather than my version which is method A. The latter requires some implicit differentiation and the substitution at the end is slightly harder. Can't see the advantage of method B if I'm honest?

 

[Deleted member 4993]

Hi Dan,
Yes I can see that when I take the question through to the final answer, I do indeed get the same answer as the book using both methods.
I am puzzled though as to why the book chooses method B rather than my version which is method A. The latter requires some implicit differentiation and the substitution at the end is slightly harder. Can't see the advantage of method B if I'm honest?

Only advantage I can tell (according to my UG professor)

extra algebra builds character!!​

It really does (and makes way for many mistakes!!)

 

[lex]

@Skelly4444
At a guess I suspect they made a mistake, copying the [imath]u^2[/imath] from the previous question.

I would also guess that this may well have been an adaptation of the Edexcel January 2009 C4 question:

 

[Skelly4444]

I wasn't sure about the extra 2 that came from my calculation and whether or not we could just add it to the constant of integration and treat it as a single entity. Am I correct in assuming that both answers are correct then?

The thing that puzzles me is why would you go to the bother of using polynomial division to obtain the quotient and remainder when it can just be integrated as it stands using the substitution I've outlined?

 

[AvgStudent]

I wasn't sure about the extra 2 that came from my calculation and whether or not we could just add it to the constant of integration and treat it as a single entity. Am I correct in assuming that both answers are correct then?

The thing that puzzles me is why would you go to the bother of using polynomial division to obtain the quotient and remainder when it can just be integrated as it stands using the substitution I've outlined?

 

[BigBeachBanana]

I wasn't sure about the extra 2 that came from my calculation and whether or not we could just add it to the constant of integration and treat it as a single entity. Am I correct in assuming that both answers are correct then?

The thing that puzzles me is why would you go to the bother of using polynomial division to obtain the quotient and remainder when it can just be integrated as it stands using the substitution I've outlined?

I think the question you should be asking is "How many other ways can I do it?"

 

[lex]

I wasn't sure about the extra 2 that came from my calculation and whether or not we could just add it to the constant of integration and treat it as a single entity. Am I correct in assuming that both answers are correct then?

The thing that puzzles me is why would you go to the bother of using polynomial division to obtain the quotient and remainder when it can just be integrated as it stands using the substitution I've outlined?

I'm wondering did you mean to post this in your other thread?

Integration of improper fraction using substitution

Having trouble agreeing with the book on this one too. Not quite sure I'm dealing with the quotient correctly. I've attached my workings which hopefully will shed some light on it.

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