Puzzle about integration by parts (H2 maths)

Mad Hat SG

I came across this puzzle while considering the integral ∫¹/_{(x lnx)} dx.

This integral has the form ^{f '(x)}/_f(x), where f(x) = ln x and f '(x) = ¹/_x. So the answer is just ln(ln x) + c. It's quite straightforward if we do it this way.

But we can also try integration by parts. But unfortunately we end up with a nonsensical answer.

We are integrating ∫ ¹/_x·¹/_{ln x} dx

Let u = ¹/_{ln x} and ^dv/_dx = ¹/_x. Then ^du/_dx = −¹/_(x(ln
x)²) and v = lnx.

The formula is ∫ u ^dv/_dx dx = uv − ∫ v ^du/_dx dx

⇒ ∫ ¹/_x·¹/_ln
x dx = 1 + ∫ ¹/_x·¹/_{ln x} dx

⇒ 0 = 1.

What does this mean and why doesn't integration by parts work in this case? There is some discussion of this on the web but I thought it is more rewarding if we figure out ourselves. Any ideas?

wee_ws SG

Hi,

From Wolfram, IBP may fail "because it leads back to the original integral" and one example has been quoted by Apostol 1967, p. 219. Please see:

http://mathworld.wolfram.com/IntegrationbyParts.html

In another source, the reason is not so straightforward, please see page 259 of the book:

http://books.google.com.sg/books?id=-yQXwhE6iWMC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

Here's another where an example is given that illustrates the non-existence of integrals even though the functions are differentiable:

http://faculty.uml.edu/jpropp/parts.pdf

Thanks!

P.S. We can also find examples where integration by substitution does not work because it leads back to the original integral. In fact, one student raised this with me today during a lesson and I'd thought of this query on the forum :)

Cheers,
Wen Shih

Mad Hat SG

Thanks for those references. Still looking at them. Some are easy to understand but one or two are a bit too cheem for me. But I look a bit more later ... no time these few days!

The issue of "leading back to the original integral" however is not a problem in this case because the equation can still proceed. Leading back to the original integral is a problem only if you get back to square one, but, in this case, that doesn't happen.

There's another famous example of this with ∫ e^xsinx dx, isn't there? If you integrate this by parts, you will eventually find yourself back with the original integral. But the equation can still proceed and you can still find the correct eventual answer.

So I think the issue must be something else. I guess this is beyond A-levels, but just for interest and understanding, I am curious ... I had never seen an example before where integration by parts leads to nonsense!

wee_ws SG

Dear Mad Hat,

Let's revisit your example where we wish to find ∫ ¹/_x·¹/_{ln x} dx

Let u = ¹/_{ln x} and ^dv/_dx = ¹/_x. Then ^du/_dx = −¹/_(x(ln
x)²) and v = lnx.

The formula is ∫ u ^dv/_dx dx = uv − ∫ v ^du/_dx dx

⇒ ∫ ¹/_x·¹/_ln
x dx = 1 + ∫ ¹/_x·¹/_{ln x} dx

⇒ 0 = 1.

We have to be careful that when we integrate 1/x, it should give ln |x|, so that

v = ln |x|,

uv = (ln |x|) / (ln x)

rather than 1.

Put simply, IBP may not work because

1. it leads back to the same integral (here we exclude the e^x . sin x type of cases),

2. the function(s) may not be integrable,

3. the function(s) may not have nice graphical behaviours.

Thanks!

Cheers,
Wen Shih

M the name SG

Hi Madhat, are you using bbcode to write the above formula or just copy and paste it?

And you might like to visit this interesting local maths site which cover topic from Kindergarten 1 to Secondary 4 :

• http://sg.ixl.com

• http://www.solvemyquestion.com.sg

Mad Hat SG

Hi Madhat, are you using bbcode to write the above formula or just copy and paste it?

And you might like to visit this interesting local maths site which cover topic from Kindergarten 1 to Secondary 4 : http://sg.ixl.com

No, I just use normal html code. It's quite hard to format. I wish I could use latex or something, then very easy. Ya, thanks for showing that site. It looks really well-designed and user-friendly. But it looks like an ang-moh site trying to capture the sg market. (Like gumtree.) I would love to come up with something like that for H2 maths, but I don't seem to have the time!

Mad Hat SG

Dear Mad Hat,

Let's revisit your example where we wish to find ∫ ¹/_x·¹/_{ln x} dx

Let u = ¹/_{ln x} and ^dv/_dx = ¹/_x. Then ^du/_dx = −¹/_(x(ln
x)²) and v = lnx.

The formula is ∫ u ^dv/_dx dx = uv − ∫ v ^du/_dx dx

⇒ ∫ ¹/_x·¹/_ln
x dx = 1 + ∫ ¹/_x·¹/_{ln x} dx

⇒ 0 = 1.

We have to be careful that when we integrate 1/x, it should give ln |x|, so that

v = ln |x|,

uv = (ln |x|) / (ln x)

rather than 1.

Put simply, IBP may not work because

1. it leads back to the same integral (here we exclude the e^x . sin x type of cases),

2. the function(s) may not be integrable,

3. the function(s) may not have nice graphical behaviours.

Thanks!

Cheers,
Wen Shih

Ya WS, thanks for that insight. I was worried a bit about this modulus issue but didn't think very hard about it. Looks now like it might be a problem! Also this issue about whether the function is differentiable/integrable everywhere is another problem! It's quite deep for me man ... my math education is not that deep, as you know. But I try!

M the name SG

If you are interested in more maths puzzle, you can visit science 2.0 : http://www.science20.com/mathematics

wee_ws SG

Dear Mad Hat,

Your deep sense of curiosity is good enough :) Have a great day ahead!

Cheers,
Wen Shih

Chemguide7 SG

Mad Hat,

Maybe you can try NUS Pure Math Analysis and Algebra modules on your own or take some uni modules from maybe Coursera. Very fun, but cannot really make money from it.

Mad Hat SG

Hey guys thanks for those suggestions. I'm not that serious in maths actually. Frankly, the subject scares me because it seems infinitely deep I just get whatever kick out of it, as and when!

By the way, sometimes when I try to login to this forum, I get this scary-looking warning. Is it serious? Do you get it once in a while also?

M the name SG

Maths Puzzle : Connect The Towns : http://www.boreme.com/posting.php?id=30207

wee_ws SG

Hi M,

Thanks for sharing this interesting content :)

Here's a paper to share:

http://www.parabola.unsw.edu.au/vol20_no2/vol20_no2_1.pdf

Enjoy!

Cheers,
Wen Shih