# S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
 It is currently Wed, 19 Jun 2013 18:45:17 UTC

 All times are UTC [ DST ]

 Page 1 of 1 [ 11 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: A strange integralPosted: Sat, 19 Jul 2003 17:07:20 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7962
Location: Lexington, MA
Hello, everyone!

Try this integral:

So far, I've found FOUR ways to integrate it.
And the four answers all look different.
(Proving them equivalent is the second phase of the problem.)

Give it a try.
I dare you to come up with a fifth method!

Top

 Post subject: Posted: Sat, 19 Jul 2003 19:54:11 UTC
 S.O.S. Oldtimer

Joined: Mon, 19 May 2003 18:41:04 UTC
Posts: 210
Location: Canada
Heres a method i got but considering you found FOUR and this one popped into my head first thing...you probably already found it:
(didnt learn latex code yet so bear with me..)

First notice:
d[ln(1+e^x)] / dx = e^x / (1 + e^x)
which can be conveniently broken into:

d[ln(1+e^x)] / dx = 1 - 1 / (1 + e^x)
Integrating both sides:

ln(1 + e^x) = int(1 - 1 / (1 + e^x)) dx
= int(dx) - int(dx / (1 + e^x))
= x - int(dx / (1 + e^x))
Rearranging:
int(dx / (1 + e^x)) = x - ln(1 + e^x) + C cant forget that constant

Top

 Post subject: a strange integralPosted: Sat, 19 Jul 2003 20:53:06 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7962
Location: Lexington, MA
Hello, radagast!

Wow! ----- Method number FIVE!

I love the deviousness of your approach: "We note that ..."
... and the fact that you never integrated it.

I thought my Method #4 was quite clever and bizarre.
But you win -- hands down!

Thank you!

Top

 Post subject: Don't look - Method #1Posted: Sun, 20 Jul 2003 14:00:33 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7962
Location: Lexington, MA
I see that there many read this thread, but few reply.

Assuming that some are still working on a response
(and others couldn't possibly care less),
I'll post my four solutions one at a time.

If you're still trying, . . . don't read any further!

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Method #1: "Long Division"

Then . . . Solution #1

Last edited by Soroban on Sun, 20 Jul 2003 15:28:53 UTC, edited 2 times in total.

Top

 Post subject: Don't look --- Method #2Posted: Sun, 20 Jul 2003 14:17:47 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7962
Location: Lexington, MA

If you don't want any hints, don't read further!

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Method #2:

Let

Multiply numerator and denominator by

Then . . . Solution #2

Last edited by Soroban on Sun, 20 Jul 2003 15:27:18 UTC, edited 2 times in total.

Top

 Post subject: Don't look --- Method #3Posted: Sun, 20 Jul 2003 15:18:11 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7962
Location: Lexington, MA

If you don't want any hints, don't read further!

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Method #3: Substitution

We have:

Let

Then

Substituting:

Now, we have two choices for completing the integration . . .

(A) Partial Fractions

(B) Complete the Square

. . . Solution #3

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Note: the substitution leads to the same ntegral.

Last edited by Soroban on Sun, 20 Jul 2003 16:49:44 UTC, edited 1 time in total.

Top

 Post subject: Don't look --- Method #4Posted: Sun, 20 Jul 2003 16:45:36 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7962
Location: Lexington, MA

If you don't want any hints, don't read any further!

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Method #4: Trig Substitution

We have:

Let

Then the denominator is:

Also,

Substituting, we get:

Since , then angle is in a right triangle with "opp" = , "adj" = 1,
and "hyp" = . Hence,

Therefore: . . . Solution #4

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

I had exhausted my imagination at this point.
(Amazing what the brain can do when it has way too much free time!)

Then Radagast surprised me with a stunning fifth method.

Is there a sixth . . . or more?'

[And don't forget to verify that solutions #1, 2, 3, 4 are equivalent!]

Top

 Post subject: Posted: Sun, 20 Jul 2003 21:48:02 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Sun, 22 Jun 2003 18:00:06 UTC
Posts: 3461
Location: North TX
let u^2 = e^x
2u*du = e^x*dx
2u*du = u^2*dx
2/u*du = dx

INT[1/(1+e^x)dx] =
INT[1/(1+u^2)*2/u*du]
INT[2/[u(1+u^2)*du] =
INT[2/u*du] - INT[2u/(1+u^2)*du] =
2ln|u| - ln|1+u^2| + C =
ln[u^2/(1+u^2)] + C =
ln[e^x/(1+e^x)] + C

Top

 Post subject: Meethod #6Posted: Mon, 21 Jul 2003 00:42:33 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7962
Location: Lexington, MA
Well done, skeeter!
And thank you!

Top

 Post subject: Posted: Mon, 21 Jul 2003 10:22:21 UTC
 S.O.S. Oldtimer

Joined: Fri, 23 May 2003 11:09:48 UTC
Posts: 206
heheh gw
Soroban, you sound pretty bored man... Maybe you wanna solve a quartic for using the quartic formula Mother Goose posted?

here's one of the top of my head.

2x^4 - 3x^3 + x^2 - 8x - 14
have a shot, showing all working

mwhahahahahar.

j/k........or am I?

_________________
Mother Goose wrote:
When you die and go to heaven, someone asks you: "Hey dcl, you spent all those years learning about them pretty numbers, did you find a use for them eventually?"

"yeah, sure I did, I used calculus to cut out a mousepad out of a teflon baking pan"

Top

 Post subject: Posted: Wed, 23 Jul 2003 02:49:24 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Fri, 2 May 2003 14:09:58 UTC
Posts: 2482
Location: In the pub, teaching Corneo how to pick up chicks...
dcl wrote:
heheh gw
Soroban, you sound pretty bored man... Maybe you wanna solve a quartic for using the quartic formula Mother Goose posted?

here's one of the top of my head.

2x^4 - 3x^3 + x^2 - 8x - 14
have a shot, showing all working

mwhahahahahar.

j/k........or am I?

I don't htink he's that bored. But it would be pretty neat to solve a quartic using the formula using a grease pencil on his windows His neighbours would think he'd be going nuts

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 11 posts ]

 All times are UTC [ DST ]

#### Who is online

Users browsing this forum: No registered users

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forum

Search for:
 Jump to:  Select a forum ------------------ High School and College Mathematics    Algebra    Geometry and Trigonometry    Calculus    Matrix Algebra    Differential Equations    Probability and Statistics    Proposed Problems Applications    Physics, Chemistry, Engineering, etc.    Computer Science    Math for Business and Economics Advanced Mathematics    Foundations    Algebra and Number Theory    Analysis and Topology    Applied Mathematics    Other Topics in Advanced Mathematics Other Topics    Administrator Announcements    Comments and Suggestions for S.O.S. Math    Posting Math Formulas with LaTeX    Miscellaneous
Contact Us | S.O.S. Mathematics Homepage
Privacy Statement | Search the "old" CyberBoard

users online during the last hour
Powered by phpBB © 2001, 2005-2011 phpBB Group.
Copyright © 1999-2013 MathMedics, LLC. All rights reserved.
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA