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Which sets of limits to choose when there is a ± sqrt (integration by substitution)?

A part of an integration by substitution question I have is finding the limits, as it is a definite integral. However I'm confused, as the substitution given by the question is:
x = (u-4)² + 1
Which I rearranged to get
u = 4 ± sqrt(x-1)
Since there is the ±, I don't get how you would know which limits to take (as the original limits are 5 and 2, so you can get either 6 and 5 or 2 and 3 from the equation). Apparently 6 and 5 is the right answer but this doesn't really explain anything!
Reply 1
Avatar for RichE
RichE
15
Original post by Yemto
A part of an integration by substitution question I have is finding the limits, as it is a definite integral. However I'm confused, as the substitution given by the question is:
x = (u-4)² + 1
Which I rearranged to get
u = 4 ± sqrt(x-1)
Since there is the ±, I don't get how you would know which limits to take (as the original limits are 5 and 2, so you can get either 6 and 5 or 2 and 3 from the equation). Apparently 6 and 5 is the right answer but this doesn't really explain anything!

Have you tried using 2 and 3 instead - my guess is that they will work also. What you shouldn't do is mix and match (6 and 3 say).
Reply 2
Avatar for Yemto
Yemto
OP
8
Original post by RichE
Have you tried using 2 and 3 instead - my guess is that they will work also. What you shouldn't do is mix and match (6 and 3 say).

I did try that, but it didn't give the same answer for some reason. It was "2 - 8x^-1" that needed to be integrated. My calculator gives this as 1.24 with 2 on top and 3 on bottom, or -1.24 with 3 on top and 2 on bottom. However when using 6 and 5, it gives 0.54. Really confused...
Original post by Yemto
A part of an integration by substitution question I have is finding the limits, as it is a definite integral. However I'm confused, as the substitution given by the question is:
x = (u-4)² + 1
Which I rearranged to get
u = 4 ± sqrt(x-1)
Since there is the ±, I don't get how you would know which limits to take (as the original limits are 5 and 2, so you can get either 6 and 5 or 2 and 3 from the equation). Apparently 6 and 5 is the right answer but this doesn't really explain anything!


What is the function to be integrated?
Reply 4
Avatar for Yemto
Yemto
OP
8
Original post by RDKGames
What is the function to be integrated?

Integral of (2 - 8u^-1)du
Original post by Yemto
Integral of (2 - 8u^-1)du

No, what is the original integral (before substittuion)?
Reply 6
Avatar for Yemto
Yemto
OP
8
Original post by DFranklin
No, what is the original integral (before substittuion)?

Oh sorry, it was 1 / (4 + sqrt(x-1)) dx. Here's the original question if it helps too, it's part c:
https://q5g5d8m9.stackpathcdn.com/wp-content/uploads/2014/02/q7-c4-january-2011-edexcel.png
Original post by Yemto
Oh sorry, it was 1 / (4 + sqrt(x-1)) dx. Here's the original question if it helps too, it's part c:
https://q5g5d8m9.stackpathcdn.com/wp-content/uploads/2014/02/q7-c4-january-2011-edexcel.png

At a fairly casual glance I'm pretty sure you're getting issues because of the square root, and that if u < 4 you need to consider (u−4)2\sqrt{(u-4)^2} as 4−u4-u rather than the obvious u - 4.

It's the kind of thing that's easy to get wrong, and I don't want to be definitive about this when I'm shooting from the cuff, but it's what immediately springs to mind.
Reply 8
Avatar for Notnek
Notnek
21
Original post by DFranklin
At a fairly casual glance I'm pretty sure you're getting issues because of the square root, and that if u < 4 you need to consider (u−4)2\sqrt{(u-4)^2} as 4−u4-u rather than the obvious u - 4.

It's the kind of thing that's easy to get wrong, and I don't want to be definitive about this when I'm shooting from the cuff, but it's what immediately springs to mind.

The OP is asking what limits to use so isn't it just the case that you choose u=4+x−1u=4+\sqrt{x-1} since that's the only way that the denominator will be uu?

It's a sub of x=(u−4)2+1x=(u-4)^2+1 while assuming u=4+x−1u=4+\sqrt{x-1}.
Original post by Notnek
The OP is asking what limits to use so isn't it just the case that you choose u=4+x−1u=4+\sqrt{x-1} since that's the only way that the denominator will be uu?

It's a sub of x=(u−4)2+1x=(u-4)^2+1 while assuming u=4+x−1u=4+\sqrt{x-1}.

Possibly that's a better (less confusing) way of looking at it; I was trusting RichE that you "should" get the same result taking 2, 3 as the limits and guessing the most likely reason why you don't. But having a clear 1-1 transformation is always a good idea when integrating by substitution and avoids the problems here.
Reply 10
Avatar for Yemto
Yemto
OP
8
Original post by DFranklin
Possibly that's a better (less confusing) way of looking at it; I was trusting RichE that you "should" get the same result taking 2, 3 as the limits and guessing the most likely reason why you don't. But having a clear 1-1 transformation is always a good idea when integrating by substitution and avoids the problems here.

Is what you're saying that since square rooting something gives 2 solutions, sqrt((u-4)²) can give either a positive (u-4) or negative (4-u) solution, just like if it were something more simple inside the square root? I never would've considered that :smile:
Original post by Yemto
Is what you're saying that since square rooting something gives 2 solutions, sqrt((u-4)²) can give either a positive (u-4) or negative (4-u) solution, just like if it were something more simple inside the square root? I never would've considered that :smile:

Yes it can; I think what Notnek said about "make a proper choice for the square root relating x and u and you don't have to worry about this" is a better approach, but if you do use a substitution like x = (u-4)^2 + 1 where there are different choices for u then you need to be more careful.

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